5cd90f5826
until around 7fe6881 the way to define well-founded recursions was to
specify a `WellFoundedRelation` on the argument explicitly. This was
rather low-level, for example one had to predict the packing of multiple
arguments into `PProd`s, the packing of mutual functions into `PSum`s,
and the cliques that were calculated.
Then the current `termination_by` syntax was introduced, where you
specify the termination argument at a higher level (one clause per
functions, unpacked arguments), and the `WellFoundedRelation` is found
using type class resolution.
The old syntax was kept around as `termination_by'`. This is not used
anywhere in the lean, std, mathlib or the theorem-proving-in-lean
repositories,
and three occurrences I found in the wild can do without
In particular, it should be possible to express anything that the old
syntax
supported also with the new one, possibly requiring a helper type with a
suitable instance, or the following generic wrapper that now lives in
std
```
def wrap {α : Sort u} {r : α → α → Prop} (h : WellFounded r) (x : α) : {x : α // Acc r x}
```
Since the old syntax is unused, has an unhelpful name and relies on
internals, this removes the support. Now is a good time before the
refactoring that's planned in #2921.
The test suite was updated without particular surprises.
The parametric `terminationHint` parser is gone, which means we can
match on syntax more easily now, in `expandDecreasingBy?`.
25 lines
563 B
Text
25 lines
563 B
Text
mutual
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def isEven : Nat → Bool
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| 0 => true
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| n+1 => isOdd n
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def isOdd : Nat → Bool
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| 0 => false
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| n+1 => isEven n
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end
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decreasing_by apply Nat.lt_succ_self
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theorem isEven_double (x : Nat) : isEven (2 * x) = true := by
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induction x with
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| zero => simp [isEven]
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| succ x ih =>
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unfold isEven
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trace_state
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rw [Nat.mul_succ, Nat.add_succ]
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simp
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unfold isOdd
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trace_state
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simp
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exact ih
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theorem isEven_succ_succ (x : Nat) : isEven (x + 2) = isEven x := by
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conv => lhs; unfold isEven; simp; unfold isOdd; simp
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