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
2.3 KiB
Text
87 lines
2.3 KiB
Text
set_option warn.sorry false
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set_option pp.proofs true
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-- set_option trace.split.failure true
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-- set_option trace.split.debug true
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/--
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error: Tactic `split` failed: Could not split an `if` or `match` expression in the goal
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Hint: Use `set_option trace.split.failure true` to display additional diagnostic information
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n : Nat
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⊢ Fin.last n =
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match id n with
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| 0 => Fin.last 0
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| n.succ => Fin.last (n + 1)
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-/
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#guard_msgs(pass trace, all) in
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example (n : Nat) : Fin.last n = match (motive := ∀ n, Fin (n+1)) id n with
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| 0 => Fin.last 0
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| n + 1 => Fin.last (n + 1) := by
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split <;> rfl
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-- This is the type-incorrect target after generalization
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/--
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error: Type mismatch
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match n with
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| 0 => Fin.last 0
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| n.succ => Fin.last (n + 1)
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has type
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Fin (n + 1)
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but is expected to have type
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Fin (n0 + 1)
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---
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error: (kernel) declaration has metavariables '_example'
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-/
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#guard_msgs in
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example (n0 n : Nat) (h : id n0 = n) :
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Fin.last n0 =
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match (generalizing := false) (motive := ∀ n, Fin (n + 1)) n with
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| 0 => Fin.last 0
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| .succ n => Fin.last (n + 1) := sorry
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-- Maybe split could use `ndrec` to cast the result type?
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-- But not sure if the result is useful
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example (n0 n : Nat) (h : id n0 = n) :
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Fin.last n0 =
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h.symm.ndrec (motive := fun n => Fin (n + 1))
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(match (generalizing := false) (motive := ∀ n, Fin (n + 1)) n with
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| 0 => Fin.last 0
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| .succ n => Fin.last (n + 1)) := by
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split
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· sorry
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· sorry
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-- Variant with proof-valued discriminant. This works (and always has):
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example (n : Nat) (h : n > 0): Fin.last n = match (motive := ∀ n _, Fin (n+1)) n, h with
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| 0, h => by contradiction
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| n + 1, _ => Fin.last (n + 1) := by
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split
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· contradiction
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· rfl
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-- This failed, non-FVar discr.
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-- Succeeds now
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#guard_msgs(pass trace, all) in
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example (n : Nat) (hpos : n > 0): Fin.last n = match (motive := ∀ n _, Fin (n+1)) n, id hpos with
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| 0, hpos0 => by contradiction
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| n + 1, _ => Fin.last (n + 1) := by
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split
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· contradiction
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· rfl
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-- It essentially manually abstracted the discr
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example (n : Nat) (h : n > 0): Fin.last n = match (motive := ∀ n _, Fin (n+1)) n, id h with
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| 0, h => by contradiction
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| n + 1, _ => Fin.last (n + 1) := by
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generalize (id h) = h'
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split
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· contradiction
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· rfl
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