32dcc6eb89
this makes `termination_by?` even slicker. The heuristics is agressive in the non-mutual case (will omit `sizeOf` if the argument is non-dependent and the `WellFoundedRelation` relation is via `sizeOfWFRel`. In the mutual case we'd also have to check the arguments, as they line up in the termination argument, have the same types. I did not bother at this point; in the mutual case we omit `sizeOf` only if the argument type is `Nat`. As a drive-by fix, `termination_by?` now also works on functions that have only one plausible measure.
230 lines
7.1 KiB
Text
230 lines
7.1 KiB
Text
/-!
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This files tests Lean's ability to guess the right lexicographic order.
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When writing tests for GuesLex, keep in mind that it doesn't do anything
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when there is only one plausible measure (one function, only one varying argument).
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-/
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set_option showInferredTerminationBy true
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def ackermann (n m : Nat) := match n, m with
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| 0, m => m + 1
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| .succ n, 0 => ackermann n 1
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| .succ n, .succ m => ackermann n (ackermann (n + 1) m)
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def ackermann2 (n m : Nat) := match n, m with
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| m, 0 => m + 1
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| 0, .succ n => ackermann2 1 n
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| .succ m, .succ n => ackermann2 (ackermann2 m (n + 1)) n
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def ackermannList (n m : List Unit) := match n, m with
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| [], m => () :: m
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| ()::n, [] => ackermannList n [()]
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| ()::n, ()::m => ackermannList n (ackermannList (()::n) m)
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def foo2 : Nat → Nat → Nat
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| .succ n, 1 => foo2 n 1
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| .succ n, 2 => foo2 (.succ n) 1
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| n, 3 => foo2 (.succ n) 2
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| .succ n, 4 => foo2 (if n > 10 then n else .succ n) 3
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| n, 5 => foo2 (n - 1) 4
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| n, .succ m => foo2 n m
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| _, _ => 0
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mutual
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def even : Nat → Bool
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| 0 => true
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| .succ n => not (odd n)
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def odd : Nat → Bool
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| 0 => false
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| .succ n => not (even n)
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end
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mutual
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def evenWithFixed (m : String) : Nat → Bool
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| 0 => true
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| .succ n => not (oddWithFixed m n)
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def oddWithFixed (m : String) : Nat → Bool
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| 0 => false
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| .succ n => not (evenWithFixed m n)
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end
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def ping (n : Nat) := pong n
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where pong : Nat → Nat
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| 0 => 0
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| .succ n => ping n
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def hasForbiddenArg (n : Nat) (_h : n = n) (m : Nat) : Nat :=
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match n, m with
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| 0, 0 => 0
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| .succ m, n => hasForbiddenArg m rfl n
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| m, .succ n => hasForbiddenArg (.succ m) rfl n
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/-!
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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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-/
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def blowup : Nat → Nat → Nat → Nat → Nat → Nat → Nat → Nat → Nat
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| 0, 0, 0, 0, 0, 0, 0, 0 => 0
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| 0, 0, 0, 0, 0, 0, 0, .succ i => .succ (blowup i i i i i i i i)
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| 0, 0, 0, 0, 0, 0, .succ h, i => .succ (blowup h h h h h h h i)
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| 0, 0, 0, 0, 0, .succ g, h, i => .succ (blowup g g g g g g h i)
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| 0, 0, 0, 0, .succ f, g, h, i => .succ (blowup f f f f f g h i)
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| 0, 0, 0, .succ e, f, g, h, i => .succ (blowup e e e e f g h i)
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| 0, 0, .succ d, e, f, g, h, i => .succ (blowup d d d e f g h i)
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| 0, .succ c, d, e, f, g, h, i => .succ (blowup c c d e f g h i)
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| .succ b, c, d, e, f, g, h, i => .succ (blowup b c d e f g h i)
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-- Let’s try to confuse the lexicographic guessing function's
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-- unpacking of packed n-ary arguments
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def confuseLex1 : Nat → @PSigma Nat (fun _ => Nat) → Nat
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| 0, _p => 0
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| .succ n, ⟨x,y⟩ => confuseLex1 n ⟨x, .succ y⟩
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def confuseLex2 : @PSigma Nat (fun _ => Nat) → Nat
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| ⟨_,0⟩ => 0
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| ⟨0,_⟩ => 0
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| ⟨.succ y,.succ n⟩ => confuseLex2 ⟨y,n⟩
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-- NB: uses sizeOf to make the termination argument non-dependent
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def dependent : (n : Nat) → (m : Fin n) → Nat
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| 0, i => Fin.elim0 i
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| .succ 0, 0 => 0
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| .succ (.succ n), 0 => dependent (.succ n) ⟨n, n.lt_succ_self⟩
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| .succ (.succ n), ⟨.succ m, h⟩ =>
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dependent (.succ (.succ n)) ⟨m, Nat.lt_of_le_of_lt (Nat.le_succ _) h⟩
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-- NB: does not use sizeOf, as parameters in the fixed prefix are fine.
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def dependentWithFixedPrefix (n : Nat) : (m : Fin n) → (acc : Nat) → Nat
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| ⟨0, _⟩, acc => acc
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| ⟨i+1, h⟩, acc => dependentWithFixedPrefix n ⟨i, Nat.lt_of_succ_lt h⟩ (acc + i)
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-- An example based on a real world problem, condensed by Leo
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inductive Expr where
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| add (a b : Expr)
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| val (n : Nat)
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mutual
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def eval (a : Expr) : Nat :=
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match a with
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| .add x y => eval_add (x, y)
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| .val n => n
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def eval_add (a : Expr × Expr) : Nat :=
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match a with
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| (x, y) => eval x + eval y
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end
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namespace VarNames
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/-! Test that varnames are inferred nicely. -/
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def shadow1 (x2 : Nat) : Nat → Nat
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| 0 => 0
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| .succ n => shadow1 (x2 + 1) n
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decreasing_by decreasing_tactic
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-- This test is a bit moot since #3081, but lets keep it
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def some_n : Nat := 1
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def shadow2 (some_n : Nat) : Nat → Nat
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| 0 => 0
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| .succ n => shadow2 (some_n + 1) n
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decreasing_by decreasing_tactic
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-- The following test whether `sizeOf` is properly printed, and possibly qualified
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-- For this we need a type that needs an explicit “sizeOf”.
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structure OddNat where nat : Nat
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instance : WellFoundedRelation OddNat := measure (fun ⟨n⟩ => n+1)
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-- Just to check that sizeOf is actually used
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def oddNat : OddNat → Nat
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| ⟨0⟩ => 0
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| ⟨.succ n⟩ => oddNat ⟨n⟩
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decreasing_by decreasing_tactic
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-- Shadowing `sizeOf`, as a varying paramter
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def shadowSizeOf1 (sizeOf : Nat) : OddNat → Nat
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| ⟨0⟩ => 0
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| ⟨.succ n⟩ => shadowSizeOf1 (sizeOf + 1) ⟨n⟩
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decreasing_by decreasing_tactic
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-- Shadowing `sizeOf`, as a fixed paramter
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def shadowSizeOf2 (sizeOf : Nat) : OddNat → Nat → Nat
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| ⟨0⟩, m => m
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| ⟨.succ n⟩, m => shadowSizeOf2 sizeOf ⟨n⟩ m
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decreasing_by decreasing_tactic
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-- Shadowing `sizeOf`, as something in the environment
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def sizeOf : Nat := 2
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def qualifiedSizeOf (m : Nat) : OddNat → Nat
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| ⟨0⟩ => 0
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| ⟨.succ n⟩ => qualifiedSizeOf (m + 1) ⟨n⟩
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decreasing_by decreasing_tactic
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end VarNames
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namespace MutualNotNat1
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-- A type that isn't Nat, checking that the inferred argument uses `sizeOf` so that
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-- the types of the termination argument aligns.
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structure OddNat2 where nat : Nat
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instance : SizeOf OddNat2 := ⟨fun n => n.nat⟩
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@[simp] theorem OddNat2.sizeOf_eq (n : OddNat2) : sizeOf n = n.nat := rfl
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mutual
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def foo : Nat → Nat
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| 0 => 0
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| n+1 => bar ⟨n⟩
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def bar : OddNat2 → Nat
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| ⟨0⟩ => 0
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| ⟨n+1⟩ => foo n
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end
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end MutualNotNat1
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namespace MutualNotNat2
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-- A type that is defeq to Nat, but with a different `sizeOf`, checking that the
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-- inferred argument uses `sizeOf` so that the types of the termination argument aligns.
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def OddNat3 := Nat
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instance : SizeOf OddNat3 := ⟨fun n => 42 - @id Nat n⟩
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@[simp] theorem OddNat3.sizeOf_eq (n : OddNat3) : sizeOf n = 42 - @id Nat n := rfl
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mutual
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def foo : Nat → Nat
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| 0 => 0
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| n+1 =>
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if h : n < 42 then bar (42 - n) else 0
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-- termination_by x1 => x1
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decreasing_by simp_wf; simp [OddNat3]; omega
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def bar (o : OddNat3) : Nat := if h : @id Nat o < 41 then foo (41 - @id Nat o) else 0
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-- termination_by sizeOf o
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decreasing_by simp_wf; simp [id] at *; omega
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end
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namespace MutualNotNat2
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namespace MutualNotNat3
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-- A varant of the above, but where the type of the parameter refined to `Nat`.
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-- This tests if `GuessLex` is inferring the `SizeOf` instance based on the type of the
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-- concrete parameter/argument (wrong, but status quo), or based on the types in the function
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-- signature (correct, todo)
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def OddNat3 := Nat
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instance : SizeOf OddNat3 := ⟨fun n => 42 - @id Nat n⟩
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@[simp] theorem OddNat3.sizeOf_eq (n : OddNat3) : sizeOf n = 42 - @id Nat n := rfl
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mutual
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def foo : Nat → Nat
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| 0 => 0
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| n+1 =>
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if h : n < 42 then bar (42 - n) else 0
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-- termination_by x1 => x1
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decreasing_by simp_wf; simp [OddNat3]; omega
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def bar : OddNat3 → Nat
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| Nat.zero => 0
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| n+1 => if h : n < 41 then foo (40 - n) else 0
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-- termination_by x1 => sizeOf x1
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decreasing_by simp_wf; omega
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end
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namespace MutualNotNat3
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