c3948cba24
This PR upstreams the definition of `Vector` from Batteries, along with the basic functions.
37 lines
1.3 KiB
Text
37 lines
1.3 KiB
Text
def f : Nat → Nat → Nat
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| 0, y => y
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| x+1, y+1 => f (x-2) y
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| x+1, 0 => 0
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example : f 0 y = y :=
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rfl -- Error, it does not hold by reflexivity since the recursion is on `y`
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example : f 0 0 = 0 := rfl
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example : f 0 (y+1) = y+1 := rfl
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inductive Vector' (α : Type u) : Nat → Type u where
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| nil : Vector' α 0
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| cons : α → Vector' α n → Vector' α (n+1)
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namespace Vector'
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def insert (a: α): Fin (n+1) → Vector' α n → Vector' α (n+1)
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| ⟨0 , _⟩, xs => cons a xs
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| ⟨i+1, h⟩, cons x xs => cons x $ xs.insert a ⟨i, Nat.lt_of_succ_lt_succ h⟩
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theorem insert_at_0_eq_cons1 (a: α) (v: Vector' α n): v.insert a ⟨0, Nat.zero_lt_succ n⟩ = cons a v :=
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rfl -- Error, it does not hold by reflexivity because the recursion is on v
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example (a : α) : nil.insert a ⟨0, by simp_arith⟩ = cons a nil :=
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rfl
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example (a : α) (b : α) (bs : Vector' α n) : (cons b bs).insert a ⟨0, by simp_arith⟩ = cons a (cons b bs) :=
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rfl
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theorem insert_at_0_eq_cons2 (a: α) (v: Vector' α n): v.insert a ⟨0, Nat.zero_lt_succ n⟩ = cons a v := by
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rw [insert]
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theorem insert_at_0_eq_cons3 (a: α) (v: Vector' α n): v.insert a ⟨0, Nat.zero_lt_succ n⟩ = cons a v := by
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simp only [insert]
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end Vector'
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