47 lines
1.6 KiB
Text
47 lines
1.6 KiB
Text
def Vec (α : Type u) (n : Nat) : Type u := { a : List α // a.length = n }
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def Vec.nil : Vec α 0 := ⟨[], rfl⟩
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def Vec.cons (a : α) (as : Vec α n) : Vec α (n+1) := ⟨a :: as.val, by simp [as.property]⟩
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set_option pp.analyze false
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def Vec.casesOn
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(motive : (n : Nat) → Vec α n → Sort v)
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(n : Nat)
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(as : Vec α n)
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(nil : motive 0 Vec.nil)
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(cons : (n : Nat) → (a : α) → (as : Vec α n) → (ih : motive n as) → motive (n+1) (Vec.cons a as))
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: motive n as :=
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let rec go (n : Nat) (as : List α) (h : as.length = n) : motive n ⟨as, h⟩ :=
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match n, as, h with
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| 0, [], _ => nil
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| n+1, a::as, h =>
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have : as.length = n := by injection h; assumption
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have ih : motive n ⟨as, this⟩ := go n as this
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cons n a ⟨as, this⟩ ih
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match as with
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| ⟨as, h⟩ => go n as h
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example (n : Nat) (a : α) (as : Vec α n) : Vec.cons a (Vec.cons a as) = Vec.cons a (Vec.cons a as) := by
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induction n+2, Vec.cons a (Vec.cons a as) using Vec.casesOn
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case nil => constructor
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case cons n a as ih =>
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trace_state
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constructor
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#print "-----"
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example (n : Nat) (a : α) (as : Vec α n) : Vec.cons a (Vec.cons a as) = Vec.cons a (Vec.cons a as) := by
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cases n+2, Vec.cons a (Vec.cons a as) using Vec.casesOn
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case nil => constructor
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case cons n a as ih =>
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trace_state
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constructor
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#print "-----"
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example (n : Nat) (a : α) (as : Vec α n) : Vec.cons a (Vec.cons a as) = Vec.cons a (Vec.cons a as) := by
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cases h₁ : n+2, h₂ : Vec.cons a (Vec.cons a as) using Vec.casesOn
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case nil => constructor
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case cons n' a' as' ih =>
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trace_state
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constructor
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