116 lines
5.6 KiB
Markdown
116 lines
5.6 KiB
Markdown
Hi Lean Provers! Curious, I know that https://leanprover-community.github.io/mathlib4_docs/ is good for referencing the mathlib docs. If I want to see all of the main Lean4 syntax/constructs e.g., building types, strucs, functions, what is the recommended website? Similarly is there a cheat sheet/table for this?
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1 reply
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Jibiana Jakpor
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22 minutes ago
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Functional Programming in Lean is a great resource for learning the syntax, especially for general purpose programming: https://leanprover.github.io/functional_programming_in_lean/. It doesn’t have much info on tactics though. That’s where MIL or Theorem Proving in Lean 4 will serve you bette
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# Learning to write Lean
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[Opening a lean project in VSCODE.](https://proofassistants.stackexchange.com/questions/2760/how-does-one-create-a-lean-project-and-have-mathlib-import-work-when-not-creatin/3779#3779)
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We start of the bat recommending good sources for learning Lean 4:
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- [Theorem proving in Lean](https://leanprover.github.io/theorem_proving_in_lean4/)
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- []
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The goal will be to give an overview of useful tips for proving in Lean
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## Unicode
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For unicode use `\` backslash so that `\to` becomes an arrow.
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- `\^-1` for inverse e.g., `2⁻¹`
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- `\l` for back arrow e.g., in rewrite
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- `to` for arrow/implication
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- TODO: iff
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- `\R` for `ℝ`
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## Tactics
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tip: seems mathlib_4 documentation (& Moogle) are the best to find tactic docs.
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### Entering tactic mode
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- use `by` afte thm declaration e.g., `theorem thm_name : \forall n, n = n := by sorry` I think it works since `:=` expects you to write a function/proof term and then `by ...` enters tactic mode. `#print thm_name` TODO or something prints the proof term.
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## Intro & Intros tactic
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- TODO: intro intros
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- `intro` introduces one or more hypotheses, optionally naming and/or pattern-matching them. For each hypothesis to be introduced, the remaining main goal's target type must be a let or function type.
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- it seems it also unfolds definitions for you (and yes introduces hypothesis.)
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ref: [intro](https://leanprover-community.github.io/mathlib4_docs/Init/Tactics.html#Lean.Parser.Tactic.intro)
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- `intros` Introduces zero or more hypotheses, optionally naming them.
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- intros is equivalent to repeatedly applying intro until the goal is not an obvious candidate for intro, ...
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ref: [intros](https://leanprover-community.github.io/mathlib4_docs/Init/Tactics.html#Lean.Parser.Tactic.intros)
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### Rewrite Tactic
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My understanding is that rewrites is the substitution (rewrite a term) tactic when we have an equality.
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How tactic `rw` works:
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- `rw [t]` applies from left to right on first term wrt to equality term `t` on goal
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- rw tactic closes/discharges any goals of the form `t = t`.
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- rw [ <- t] or rw [ \l t ] to apply equality form left to right (on 1st term) on goal
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- rw to rewrite hypothesis `h` do `rw [t] at h`
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- (rw to rewrite everything I assume rw [*] but then proof harder to read!)
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- to apply tactic at specific loc do rw `[Nat.add_comm b]` if `a + (b + c)` --> `a + (c + b)`
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- tip: hover over Nat.add_comm to see how tactic and arg interact.
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- rewrite using compund expression `rw [h1 h2]` <--> `rw [h1]; rw [h2]`
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- rewrite can also rewrite terms that aren't equalities
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- e.g., if `h_k_eq_0: k = 0` then `t : Pair 1 k` --> `t : Pair 1 0` with `rw [h_k_eq_0] at t`
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ref: https://leanprover.github.io/theorem_proving_in_lean4/tactics.html
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- `rwa` calls `rw`, then closes any remaining goals using `assumption`. Note: found by writing `rwa` in tactic mode then using vscode to get to it's def. Mathlib4 search, Moogle, didn't help surprisingly.
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### Constructor tactic
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- `constructor` If the main goal's target type is an inductive type, constructor solves it with the first matching constructor, or else fails.
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-tactic introduces a certain number of new proof obligations/goals to discharge/close according to each term in the constructor of the goal.
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- e.g., if we have `... |- a \and k = 0 -> c` the constructor will open two goals where you need to prove `a` and `k = 0`. i.e., to have arrived at that goal you must have had a proof/evidence that `a` and `k=0` had proves e.g., `k = 0` might be a simple assumption in your "local context"/hypothesis space (TODO lean official lingo)
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### Have
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- `have : t := e` := "introduces theorem with proof `e` e.g., `e` can be `by tactics...` or the exact `proof term`
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e.g.,
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```lean
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have h_pos : 0 < x⁻¹ := inv_pos.mpr h_x_pos
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```
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### Dot deperator for cases
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Do `.` to handle each case e.g., in induction.
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### Tacticals
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Note: `;` is not a tactical. TODO what is?
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### Expressions
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TODO: `\forall x : R, x < 0'
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### Mathlib tips
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- use less than (lt) in terms (e.g., thms) so it's easier to prove things.
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- `m_lt_m_right` TODO: what is tip?
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### Seeing Propositional Constructors
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ref: [How do I explicitly see the propositional or logical constructors in Lean 4?](https://proofassistants.stackexchange.com/questions/3794/how-do-i-explicitly-see-the-propositional-or-logical-constructors-in-lean-4)
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TODO: precedence of exists, forall vs And, implicaiton.
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And < (less precedence than) Implcation <==> `A /\ B -> B` means `(A /\ B) -> B`.
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Exists delta, P delta -> P' delta <==>
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### Terminology
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- argument vs parameter -> argument (calling) the argument of f is 2, parameter (declaration) of f is n: Nat
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- elaboration
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- proof term
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- proof obligation
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- discharge
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### Questions:
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Q: what is a macro again?
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Q: why is there this Init.Tactics vs Std.Tactic.Rcases
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https://leanprover-community.github.io/mathlib4_docs/Std/Tactic/RCases.html#Std.Tactic.rintro
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https://leanprover-community.github.io/mathlib4_docs/Init/Tactics.html#Lean.Parser.Tactic.constructor
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Q: destructing patterns, rcases, rintro & rcases vs constructor
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