cbba783bcf
This improves Lean’s capabilities to guess the termination measure for
well-founded
recursion, by also trying lexicographic orders. For example:
def ackermann (n m : Nat) := match n, m with
| 0, m => m + 1
| .succ n, 0 => ackermann n 1
| .succ n, .succ m => ackermann n (ackermann (n + 1) m)
now just works.
The module docstring of `Lean.Elab.PreDefinition.WF.GuessLex` tells the
technical story.
Fixes #2837
51 lines
1.7 KiB
Text
51 lines
1.7 KiB
Text
/-!
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A “tricky” example from “Finding Lexicographic Orders for Termination Proofs in
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Isabelle/HOL” by Lukas Bulwahn, Alexander Krauss, and Tobias Nipkow,
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10.1007/978-3-540-74591-4_5
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At the time of writing, Lean is able to find the lexicographic order
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just fine, but only if the tactic is powerful enough. In partiuclar,
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the default `decreasing_tactic` can only handle lexicographic descend when either
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the left gets smaller, or the left stays equal and the right gets smaller.
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But here we need to allow the general form, where the left is ≤ and the right
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gets smaller. This needs a backtracking proof search, it seems, which we build here
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(`search_lex`).
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-/
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macro_rules | `(tactic| decreasing_trivial) =>
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`(tactic| apply Nat.le_refl)
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macro_rules | `(tactic| decreasing_trivial) =>
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`(tactic| apply Nat.succ_lt_succ; decreasing_trivial)
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macro_rules | `(tactic| decreasing_trivial) =>
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`(tactic| apply Nat.sub_le)
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macro_rules | `(tactic| decreasing_trivial) =>
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`(tactic| apply Nat.div_le_self)
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syntax "search_lex " tacticSeq : tactic
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macro_rules | `(tactic|search_lex $ts:tacticSeq) => `(tactic| (
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solve
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| apply Prod.Lex.right'
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· $ts
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· search_lex $ts
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| apply Prod.Lex.left
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· $ts
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| $ts
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))
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-- set_option trace.Elab.definition.wf true in
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mutual
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def prod (x y z : Nat) : Nat :=
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if y % 2 = 0 then eprod x y z else oprod x y z
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def oprod (x y z : Nat) := eprod x (y - 1) (z + x)
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def eprod (x y z : Nat) := if y = 0 then z else prod (2 * x) (y / 2) z
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end
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-- termination_by
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-- prod x y z => (y, 2)
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-- oprod x y z => (y, 1)
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-- eprod x y z => (y, 0)
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decreasing_by
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simp_wf
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search_lex solve
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| decreasing_trivial
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| apply Nat.bitwise_rec_lemma; assumption
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