f9dc77673b
This PR lets recursive functions defined by well-founded recursion use a different `fix` function when the termination measure is of type `Nat`. This fix-point operator use structural recursion on “fuel”, initialized by the given measure, and is thus reasonable to reduce, e.g. in `by decide` proofs. Extra provisions are in place that the fixpoint operator only starts reducing when the fuel is fully known, to prevent “accidential” defeqs when the remaining fuel for the recursive calls match the initial fuel for that recursive argument. To opt-out, the idiom `termination_by (n,0)` can be used. We still use `@[irreducible]` as the default for such recursive definitions, to avoid unexpected `defeq` lemmas. Making these functions `@[semireducible]` by default showed performance regressions in lean. When the measure is of type `Nat`, the system will accept an explicit `@[semireducible]` without the usual warning. Fixes #5234. Fixes: #11181.
87 lines
3.2 KiB
Text
87 lines
3.2 KiB
Text
def trailingZeros (i : Int) : Nat :=
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if h : i = 0 then 0 else aux i.natAbs i h (Nat.le_refl _) 0
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where
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aux (k : Nat) (i : Int) (hi : i ≠ 0) (hk : i.natAbs ≤ k) (acc : Nat) : Nat :=
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match k, (by omega : k ≠ 0) with
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| k + 1, _ =>
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if h : i % 2 = 0 then aux k (i / 2) (by omega) (by omega) (acc + 1)
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else acc
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termination_by structural k
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/--
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info: equations:
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@[defeq] theorem trailingZeros.aux.eq_1 : ∀ (i : Int) (hi : i ≠ 0) (acc k_2 : Nat) (x_1 : k_2 + 1 ≠ 0)
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(hk_2 : i.natAbs ≤ k_2 + 1),
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trailingZeros.aux k_2.succ i hi hk_2 acc = if h : i % 2 = 0 then trailingZeros.aux k_2 (i / 2) ⋯ ⋯ (acc + 1) else acc
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-/
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#guard_msgs(pass trace, all) in
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#print equations trailingZeros.aux
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-- set_option trace.Elab.definition.eqns true
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-- set_option trace.split.debug true
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-- set_option trace.Meta.Match.unify true
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def trailingZeros' (i : Int) : Nat :=
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if h : i = 0 then 0 else aux i.natAbs i h (Nat.le_refl _) 0
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where
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aux (k : Nat) (i : Int) (hi : i ≠ 0) (hk : i.natAbs ≤ k) (acc : Nat) : Nat :=
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match k, (by omega : k ≠ 0) with
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| k + 1, _ =>
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if h : i % 2 = 0 then aux k (i / 2) (by omega) (by omega) (acc + 1)
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else acc
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termination_by k
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/--
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info: equations:
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theorem trailingZeros'.aux.eq_1 : ∀ (i : Int) (hi : i ≠ 0) (acc k_2 : Nat) (x_1 : k_2 + 1 ≠ 0)
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(hk_2 : i.natAbs ≤ k_2 + 1),
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trailingZeros'.aux k_2.succ i hi hk_2 acc =
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if h : i % 2 = 0 then trailingZeros'.aux k_2 (i / 2) ⋯ ⋯ (acc + 1) else acc
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-/
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#guard_msgs(pass trace, all) in
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#print equations trailingZeros'.aux
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def trailingZeros2 (i : Int) : Nat :=
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if h : i = 0 then 0 else aux i.natAbs i h (Nat.le_refl _) 0
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where
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aux (k : Nat) (i : Int) (hi : i ≠ 0) (hk : i.natAbs ≤ k) (acc : Nat) : Nat :=
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match k with
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| k + 1 =>
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if h : i % 2 = 0 then aux k (i / 2) (by omega) (by omega) (acc + 1)
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else acc
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| 0 => by omega
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termination_by structural k
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/--
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info: equations:
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@[defeq] theorem trailingZeros2.aux.eq_1 : ∀ (i : Int) (hi : i ≠ 0) (acc k_2 : Nat) (hk_2 : i.natAbs ≤ k_2 + 1),
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trailingZeros2.aux k_2.succ i hi hk_2 acc =
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if h : i % 2 = 0 then trailingZeros2.aux k_2 (i / 2) ⋯ ⋯ (acc + 1) else acc
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@[defeq] theorem trailingZeros2.aux.eq_2 : ∀ (i : Int) (hi : i ≠ 0) (acc : Nat) (hk_2 : i.natAbs ≤ 0),
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trailingZeros2.aux 0 i hi hk_2 acc = acc
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-/
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#guard_msgs(pass trace, all) in
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#print equations trailingZeros2.aux
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def trailingZeros2' (i : Int) : Nat :=
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if h : i = 0 then 0 else aux i.natAbs i h (Nat.le_refl _) 0
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where
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aux (k : Nat) (i : Int) (hi : i ≠ 0) (hk : i.natAbs ≤ k) (acc : Nat) : Nat :=
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match k with
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| k + 1 =>
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if h : i % 2 = 0 then aux k (i / 2) (by omega) (by omega) (acc + 1)
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else acc
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| 0 => by omega
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termination_by k
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/--
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info: equations:
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theorem trailingZeros2'.aux.eq_1 : ∀ (i : Int) (hi : i ≠ 0) (acc k_2 : Nat) (hk_2 : i.natAbs ≤ k_2 + 1),
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trailingZeros2'.aux k_2.succ i hi hk_2 acc =
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if h : i % 2 = 0 then trailingZeros2'.aux k_2 (i / 2) ⋯ ⋯ (acc + 1) else acc
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@[defeq] theorem trailingZeros2'.aux.eq_2 : ∀ (i : Int) (hi : i ≠ 0) (acc : Nat) (hk_2 : i.natAbs ≤ 0),
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trailingZeros2'.aux 0 i hi hk_2 acc = acc
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-/
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#guard_msgs(pass trace, all) in
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#print equations trailingZeros2'.aux
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