d6c81f8594
With
set_option showInferredTerminationBy true
this prints a message like
Inferred termination argument:
termination_by
ackermann n m => (sizeOf n, sizeOf m)
it tries hard to use names that
* match the names that the user used, if present
* have no daggers (so that it can be copied)
* do not shadow each other
* do not shadow anything from the environment (just to be nice)
it does so by appending sufficient `'` to the name.
Some of the emitted `sizeOf` calls are unnecessary, but they are needed
sometimes with dependent parameters. A follow-up PR will not emit them
for non-dependent arguments, so that in most cases the output is pretty.
Somewhen down the road we also want a code action, maybe triggered by
`termination_by?`. This should come after #2921, as that simplifies that
feature (no need to merge termination arguments from different cliques
for example.)
53 lines
1.8 KiB
Text
53 lines
1.8 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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set_option showInferredTerminationBy true
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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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