24cb133eb2
This PR introduces an explicit `defeq` attribute to mark theorems that can be used by `dsimp`. The benefit of an explicit attribute over the prior logic of looking at the proof body is that we can reliably omit theorem bodies across module boundaries. It also helps with intra-file parallelism. If a theorem is syntactically defined by `:= rfl`, then the attribute is assumed and need not given explicitly. This is a purely syntactic check and can be fooled, e.g. if in the current namespace, `rfl` is not actually “the” `rfl` of `Eq`. In that case, some other syntax has be used, such as `:= (rfl)`. This is also the way to go if a theorem can be proved by `defeq`, but one does not actually want `dsimp` to use this fact. The `defeq` attribute will look at the *type* of the declaration, not the body, to check if it really holds definitionally. Because of different reduction settings, this can sometimes go wrong. Then one should also write `:= (rfl)`, if one does not want this to be a defeq theorem. (If one does then this is currently not possible, but it’s probably a bad idea anyways). The `set_option debug.tactic.simp.checkDefEqAttr true`, `dsimp` will warn if could not apply a lemma due to a missing `defeq` attribute. With `set_option backward.dsimp.useDefEqAttr.get false` one can revert to the old behavior of inferring rfl-ness based on the theorem body. Both options will go away eventually (too bad we can’t mark them as deprecated right away, see #7969) Meta programs that generate theorems (e.g. equational theorems) can use `inferDefEqAttr` to set the attribute based on the theorem body of the just created declaration. This builds on #8501 to update Init to `@[expose]` a fair amount of definitions that, if not exposed, would prevent some existing `:= rfl` theorems from being `defeq` theorems. In the interest of starting backwards compatible, I exposed these function. Hopefully many can be un-exposed later again. A mathlib adaption branch exists that includes both the meta programming fixes and changes to the theorems (e.g. changing `:= by rfl` to `:= rfl`). With the module system there is now no special handling for `defeq` theorem bodies, because we don’t look at the body anymore. The previous hack is removed. The `defeq`-ness of the theorem needs to be checked in the context of the theorem’s *type*; the error message contains a hint if the defeq check fails because of the exported context.
37 lines
1.3 KiB
Text
37 lines
1.3 KiB
Text
def f : Nat → Nat → Nat
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| 0, y => y
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| x+1, y+1 => f (x-2) y
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| x+1, 0 => 0
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example : f 0 y = y :=
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rfl -- Error, it does not hold by reflexivity since the recursion is on `y`
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example : f 0 0 = 0 := rfl
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example : f 0 (y+1) = y+1 := rfl
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inductive Vector' (α : Type u) : Nat → Type u where
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| nil : Vector' α 0
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| cons : α → Vector' α n → Vector' α (n+1)
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namespace Vector'
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def insert (a: α): Fin (n+1) → Vector' α n → Vector' α (n+1)
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| ⟨0 , _⟩, xs => cons a xs
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| ⟨i+1, h⟩, cons x xs => cons x $ xs.insert a ⟨i, Nat.lt_of_succ_lt_succ h⟩
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theorem insert_at_0_eq_cons1 (a: α) (v: Vector' α n): v.insert a ⟨0, Nat.zero_lt_succ n⟩ = cons a v :=
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(rfl) -- Error, it does not hold by reflexivity because the recursion is on v
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example (a : α) : nil.insert a ⟨0, by simp +arith⟩ = cons a nil :=
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rfl
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example (a : α) (b : α) (bs : Vector' α n) : (cons b bs).insert a ⟨0, by simp +arith⟩ = cons a (cons b bs) :=
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rfl
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theorem insert_at_0_eq_cons2 (a: α) (v: Vector' α n): v.insert a ⟨0, Nat.zero_lt_succ n⟩ = cons a v := by
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rw [insert]
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theorem insert_at_0_eq_cons3 (a: α) (v: Vector' α n): v.insert a ⟨0, Nat.zero_lt_succ n⟩ = cons a v := by
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simp only [insert]
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end Vector'
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