f20cae3729
This PR sets the `irreducible` attribute before generating the equations for recursive definitions. This prevents these equations to be marked as `defeq`, which could lead to `simp` generation proofs that do not type check at default transparency. This issue is surfacing more easily since well-founded recursion on `Nat` is implemented with a dedicated fix point operator (#7965). Before that, `WellFounded.fix` was used, which is inherently not reducing, so we did get the desired result even without the explicit reducibility setting. Fixes #12398.
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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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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