8655f7706f
This PR changes how Lean proves the equational theorems for structural recursion. The core idea is to let-bind the `f` argument to `brecOn` and rewriting `.brecOn` with an unfolding theorem. This means no extra case split for the `.rec` in `.brecOn` is needed, and `simp` doesn't change the `f` argument which can break the definitional equality with the defined function. With this, we can prove the unfolding theorem first, and derive the equational theorems from that, like for all other ways of defining recursive functions. Backs out the changes from #10415, the old strategy works well with the new goals. Fixes #5667 Fixes #10431 Fixes #10195 Fixes #2962
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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