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
78 lines
1.9 KiB
Text
78 lines
1.9 KiB
Text
inductive L (α : Type u) : Type u where
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| nil : L α
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| cons : α → L α → L α
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def L.eq [BEq α] (xs ys : L α) : Bool :=
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match _ : xs.ctorIdx == ys.ctorIdx with
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| false => false
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| true =>
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match xs, ys with
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| .nil, .nil => true
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| .cons x xs, .cons y ys => x == y && xs.eq ys
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/--
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info: theorem L.eq.eq_def.{u_1} : ∀ {α : Type u_1} [inst : BEq α] (xs ys : L α),
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xs.eq ys =
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match h : xs.ctorIdx == ys.ctorIdx with
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| false => false
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| true =>
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match xs, ys, h with
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| L.nil, L.nil, h => true
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| L.cons x xs, L.cons y ys, h => x == y && xs.eq ys
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-/
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#guard_msgs in
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#print sig L.eq.eq_def
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-- and mutual
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mutual
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inductive L1 (α : Type u) : Type u where
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| nil : L1 α
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| cons : α → L2 α → L1 α
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inductive L2 (α : Type u) : Type u where
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| nil : L2 α
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| cons : α → L1 α → L2 α
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end
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mutual
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def L1.eq [BEq α] (xs ys : L1 α) : Bool :=
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match _ : xs.ctorIdx == ys.ctorIdx with
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| false => false
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| true =>
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match xs, ys with
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| .nil, .nil => true
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| .cons x xs, .cons y ys => x == y && xs.eq ys
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def L2.eq [BEq α] (xs ys : L2 α) : Bool :=
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match _ : xs.ctorIdx == ys.ctorIdx with
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| false => false
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| true =>
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match xs, ys with
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| .nil, .nil => true
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| .cons x xs, .cons y ys => x == y && xs.eq ys
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end
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/--
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info: theorem L1.eq.eq_def.{u_1} : ∀ {α : Type u_1} [inst : BEq α] (xs ys : L1 α),
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xs.eq ys =
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match h : xs.ctorIdx == ys.ctorIdx with
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| false => false
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| true =>
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match xs, ys, h with
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| L1.nil, L1.nil, h => true
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| L1.cons x xs, L1.cons y ys, h => x == y && xs.eq ys
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-/
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#guard_msgs in
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#print sig L1.eq.eq_def
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/--
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info: theorem L2.eq.eq_def.{u_1} : ∀ {α : Type u_1} [inst : BEq α] (xs ys : L2 α),
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xs.eq ys =
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match h : xs.ctorIdx == ys.ctorIdx with
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| false => false
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| true =>
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match xs, ys, h with
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| L2.nil, L2.nil, h => true
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| L2.cons x xs, L2.cons y ys, h => x == y && xs.eq ys
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-/
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#guard_msgs in
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#print sig L2.eq.eq_def
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