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.
12 lines
253 B
Text
12 lines
253 B
Text
unfold1.lean:22:4-22:8: error: simp made no progress
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case succ
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x : Nat
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ih : isEven (2 * x) = true
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⊢ (match 2 * (x + 1) with
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| 0 => true
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| Nat.succ n => isOdd n) =
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true
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case succ
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x : Nat
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ih : isEven (2 * x) = true
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⊢ isEven (2 * x) = true
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