15.5. 终结位置与非终结位置
为了编写可维护的证明，除非 simp 能关闭目标，否则应避免不带 Lean.Parser.Tactic.simp : tacticThe `simp` tactic uses lemmas and hypotheses to simplify the main goal target or
non-dependent hypotheses. It has many variants:
- `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`.
- `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.-
- If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated
with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`.
- `simp [*]` simplifies the main goal target using the lemmas tagged with the
attribute `[simp]` and all hypotheses.
- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.
- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]`, but removes the ones named `idᵢ`.
- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If
the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis
`hᵢ` is introduced, but the old one remains in the local context.
- `simp at *` simplifies all the hypotheses and the target.
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the
other hypotheses.
only 使用它。
这种不关闭目标的 simp 用法称为 非终结 simp。
这是因为向默认 simp 集添加规则可能会增强 simp，也可能只是使它选择不同的重写序列，从而得到不同的 simp 范式。
指定 Lean.Parser.Tactic.simp : tacticThe `simp` tactic uses lemmas and hypotheses to simplify the main goal target or
non-dependent hypotheses. It has many variants:
- `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`.
- `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.-
- If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated
with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`.
- `simp [*]` simplifies the main goal target using the lemmas tagged with the
attribute `[simp]` and all hypotheses.
- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.
- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]`, but removes the ones named `idᵢ`.
- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If
the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis
`hᵢ` is introduced, but the old one remains in the local context.
- `simp at *` simplifies all the hypotheses and the target.
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the
other hypotheses.
only 后，新增引理不会影响该次策略调用。
在实践中，simp 的终结用法远不容易因新增 simp 引理而失效；即使失效，问题也更容易理解和修复。
在非终结位置工作时，可以使用 simp?（或其他名称中带有 ? 的简化策略）生成带 Lean.Parser.Tactic.simp : tacticThe `simp` tactic uses lemmas and hypotheses to simplify the main goal target or
non-dependent hypotheses. It has many variants:
- `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`.
- `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.-
- If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated
with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`.
- `simp [*]` simplifies the main goal target using the lemmas tagged with the
attribute `[simp]` and all hypotheses.
- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.
- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]`, but removes the ones named `idᵢ`.
- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If
the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis
`hᵢ` is introduced, but the old one remains in the local context.
- `simp at *` simplifies all the hypotheses and the target.
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the
other hypotheses.
only 的适当调用。
正如 apply? 或 rw? 会建议使用相关引理，simp? 会建议一次 simp 调用，其中包含达到该范式所用的最小 simp 集。
使用 simp?
此证明中的非终结 simp? 会建议一个带 Lean.Parser.Tactic.simp : tacticThe `simp` tactic uses lemmas and hypotheses to simplify the main goal target or
non-dependent hypotheses. It has many variants:
- `simp` simplifies the main goal target using lemmas tagged with the attribute `[simp]`.
- `simp [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.-
- If an `hᵢ` is a defined constant `f`, then `f` is unfolded. If `f` has equational lemmas associated
with it (and is not a projection or a `reducible` definition), these are used to rewrite with `f`.
- `simp [*]` simplifies the main goal target using the lemmas tagged with the
attribute `[simp]` and all hypotheses.
- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.
- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged
with the attribute `[simp]`, but removes the ones named `idᵢ`.
- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If
the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis
`hᵢ` is introduced, but the old one remains in the local context.
- `simp at *` simplifies all the hypotheses and the target.
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the
other hypotheses.
only、规模更小的 simp：
example (xs : Array Unit) : xs.size = 2 → xs = #[(), ()] := xs:Array Unit⊢ xs.size = 2 → xs = #[(), ()]
xs:Array Unita✝:xs.size = 2⊢ xs = #[(), ()]
xs:Array Unita✝:xs.size = 2⊢ xs.size = #[(), ()].size
xs:Array Unita✝:xs.size = 2⊢ xs.size = 2
All goals completed! 🐙
建议的改写是：
由此得到更易维护的证明：
example (xs : Array Unit) : xs.size = 2 → xs = #[(), ()] := xs:Array Unit⊢ xs.size = 2 → xs = #[(), ()]
xs:Array Unita✝:xs.size = 2⊢ xs = #[(), ()]
xs:Array Unita✝:xs.size = 2⊢ xs.size = #[(), ()].size
xs:Array Unita✝:xs.size = 2⊢ xs.size = 2
All goals completed! 🐙