0c39a50337
This PR renames `Std.List.Zipper` to `List.Cursor`, with slight changes to the implementation (no `reverse`) and use in loop specification lemmas.
127 lines
3.7 KiB
Text
127 lines
3.7 KiB
Text
/-
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This is from Markus's blog post on monadic reasoning:
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https://markushimmel.de/blog/my-first-verified-imperative-program/
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-/
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import Std.Data.HashSet.Lemmas
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import Std.Tactic.Do
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open Std Do
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/-!
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# Missing standard library material that will be part of a future Lean release
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-/
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namespace List
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theorem exists_mem_iff_exists_getElem (P : α → Prop) (l : List α) :
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(∃ x ∈ l, P x) ↔ ∃ (i : Nat), ∃ hi, P (l[i]'hi) := by
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grind [mem_iff_getElem]
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/--
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`l.ExistsPair P` asserts that there are indices `i` and `j` such that `i < j` and `P l[i] l[j]` is true.
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-/
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structure ExistsPair (P : α → α → Prop) (l : List α) where
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not_pairwise : ¬l.Pairwise (¬P · ·)
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theorem existsPair_iff_not_pairwise {P : α → α → Prop} {l : List α} :
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l.ExistsPair P ↔ ¬l.Pairwise (fun x y => ¬P x y) := by
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grind [ExistsPair]
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@[simp, grind]
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theorem not_existsPair_nil {P : α → α → Prop} : ¬[].ExistsPair P := by
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simp [existsPair_iff_not_pairwise]
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@[simp, grind]
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theorem existsPair_cons {P : α → α → Prop} {x : α} {xs : List α} :
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(x::xs).ExistsPair P ↔ (∃ y ∈ xs, P x y) ∨ xs.ExistsPair P := by
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grind [List.existsPair_iff_not_pairwise, pairwise_iff_forall_sublist]
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@[simp, grind]
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theorem existsPair_append {P : α → α → Prop} {xs ys : List α} :
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(xs ++ ys).ExistsPair P ↔ xs.ExistsPair P ∨ ys.ExistsPair P ∨ (∃ x ∈ xs, ∃ y ∈ ys, P x y) := by
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grind [List.existsPair_iff_not_pairwise]
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end List
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-- Tell Lean that it doesn't need to warn us that `mvcgen` is still a pre-release.
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set_option mvcgen.warning false
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/-!
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# Imperative implementation
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-/
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def pairsSumToZero (l : List Int) : Bool := Id.run do
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let mut seen : HashSet Int := ∅
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for x in l do
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if -x ∈ seen then
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return true
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seen := seen.insert x
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return false
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/-!
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# Verification of imperative implementation
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-/
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theorem pairsSumToZero_correct (l : List Int) : pairsSumToZero l ↔ l.ExistsPair (fun a b => a + b = 0) := by
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generalize h : pairsSumToZero l = r
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apply Id.of_wp_run_eq h
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mvcgen
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case inv1 =>
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exact Invariant.withEarlyReturn
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(onReturn := fun r b => ⌜r = true ∧ l.ExistsPair (fun a b => a + b = 0)⌝)
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(onContinue := fun traversalState seen =>
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⌜(∀ x, x ∈ seen ↔ x ∈ traversalState.prefix) ∧ ¬traversalState.prefix.ExistsPair (fun a b => a + b = 0)⌝)
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all_goals simp_all <;> grind
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/--
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trace: l : List Int
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⊢ (match
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(forIn l ⟨none, ∅⟩ fun x r =>
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if -x ∈ r.snd then pure (ForInStep.done ⟨some true, r.snd⟩)
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else pure (ForInStep.yield ⟨none, r.snd.insert x⟩)).run.fst with
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| none => pure false
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| some a => pure a).run =
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true ↔
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List.ExistsPair (fun a b => a + b = 0) l
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---
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warning: declaration uses 'sorry'
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-/
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#guard_msgs in
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theorem pairsSumToZero_correct_directly (l : List Int) : pairsSumToZero l ↔ l.ExistsPair (fun a b => a + b = 0) := by
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simp [pairsSumToZero]
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trace_state
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-- give up; need lemma about `forIn`
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admit
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namespace Functional
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/-!
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# Bonus: functional implementation
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-/
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def pairsSumToZero (l : List Int) : Bool :=
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go l ∅
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where
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go (m : List Int) (seen : HashSet Int) : Bool :=
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match m with
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| [] => false
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| x::xs => if -x ∈ seen then true else go xs (seen.insert x)
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/-!
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# Bonus: verification of functional implementation
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-/
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theorem pairsSumToZero_go_iff (l : List Int) (seen : HashSet Int) :
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pairsSumToZero.go l seen = true ↔
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l.ExistsPair (fun a b => a + b = 0) ∨ ∃ a ∈ seen, ∃ b ∈ l, a + b = 0 := by
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fun_induction pairsSumToZero.go <;> simp_all <;> grind
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theorem pairsSumToZero_iff (l : List Int) :
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pairsSumToZero l = true ↔ l.ExistsPair (fun a b => a + b = 0) := by
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simp [pairsSumToZero, pairsSumToZero_go_iff]
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