45fccc5906
Replaces `@[eliminator]` with two attributes `@[induction_eliminator]` and `@[cases_eliminator]` for defining custom eliminators for the `induction` and `cases` tactics, respectively. Adds `Nat.recAux` and `Nat.casesAuxOn`, which are eliminators that are defeq to `Nat.rec` and `Nat.casesOn`, but these use `0` and `n + 1` rather than `Nat.zero` and `Nat.succ n`. For example, using `induction` to prove that the factorial function is positive now has the following goal states (thanks also to #3616 for the goal state after unfolding). ```lean example : 0 < fact x := by induction x with | zero => decide | succ x ih => /- x : Nat ih : 0 < fact x ⊢ 0 < fact (x + 1) -/ unfold fact /- ... ⊢ 0 < (x + 1) * fact x -/ simpa using ih ``` Thanks to @adamtopaz for initial work on splitting the `@[eliminator]` attribute.
14 lines
356 B
Text
14 lines
356 B
Text
theorem n_minus_one_le_n {n : Nat} : n > 0 → n - 1 < n := by
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cases n with
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| zero => simp []
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| succ n =>
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intros
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rw [Nat.add_sub_cancel]
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apply Nat.le.refl
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partial def foo : Array Int → Int
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| arr => Id.run do
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let mut r : Int := 1
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while h : arr.size > 0 do
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r := r * arr[arr.size - 1]'(by apply n_minus_one_le_n h)
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return r
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