75d79819c3
This PR adds a focused error explanation aimed at the case where someone tries to use Natural-Numbers-Game-style `induction` proofs directly in Lean, where such proofs are not syntactically valid. ## Discussion The natural numbers game uses a syntax that overlaps with Lean's `induction` syntax despite having more structural similarity to `induction'`. This means that fully correct proofs in the natural numbers game, like this... ```lean4 import Mathlib theorem zero_mul (m : ℕ) : 0 * m = 0 := by induction m with n n_ih rw [mul_zero] rfl rw [mul_succ] rw [add_zero] rw [n_ih] rfl ``` ...have completely baffling error messages from a newcomers' perspective: ``` notNaturalNumbersGame.lean:3:20: error: unknown tactic notNaturalNumbersGame.lean:3:2: error: Alternative `zero` has not been provided notNaturalNumbersGame.lean:3:2: error: Alternative `succ` has not been provided ``` (the Mathlib import here only provides the `ℕ` syntax here; equivalently `ℕ` could be renamed to `Nat` and the import could be removed, [like this](https://live.lean-lang.org/#codez=C4Cwpg9gTmC2AEAvMUIH1YFcA28AUCAXPAHICGwAlPMQAzwBU8CAvPPYWwEYCeAUPHgBLAHYATTAGNgQiCObwA7kNDx5ItEJAD4URfADaWbGmSoAujqgAzbFf1GcaAM5TJlwXsNkxY0yggPXQcNLSCbbCA)) There are many problems with this proof from the perspective of "stock" Lean, but the error messages in the `induction` case are particularly unfriendly and provide no guidance from a NNG learner's perspective. This PR provides more information about what is wrong: ``` notNaturalNumbersGame.lean:3:20: error: unknown tactic notNaturalNumbersGame.lean:3:14: error(lean.inductionWithNoAlts): Invalid syntax for induction tactic: The `with` keyword must followed by a tactic or by an alternative (e.g. `| zero =>`), but here it is followed by the identifier `n`. ``` The error explanation it links to explicitly flags the transition of NNG-style proofs to Lean as the likely culprit, and gives an example of an effective translation.
25 lines
685 B
Text
25 lines
685 B
Text
-- Catch special case of induction pattern that uses Mathlib's alternative `induction'` syntax,
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-- which is used in the Natural Numbers game as just `induction`, overlapping with the parser for
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-- Lean's default tactic
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theorem zero_mul (m : Nat) : 0 * m = 0 := by
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induction m with n n_ih
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rw [Nat.mul_zero]
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rw [Nat.mul_succ]
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rw [Nat.add_zero]
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rw [n_ih]
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theorem zero_mul2 (m : Nat) : 0 * m = 0 := by
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cases m with n
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rw [Nat.mul_zero]
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rw [Nat.mul_succ]
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rw [Nat.add_zero]
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rw [zero_mul]
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-- Special case not triggered by empty case analysis
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theorem zero_mul3 (m : Nat) : 0 * m = 0 := by
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induction m with
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theorem zero_mul4 (m : Nat) : 0 * m = 0 := by
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cases m with
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