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(c) 2020 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nmodule\n\npublic import Mathlib.Init\npublic import Batteries.Util.LibraryNote\n\n/-!\n# Documentation for `ext` tactic\n\nThis file contains a library note on the use of the `ext` tactic and `@[ext]` attribute.\n-/\n\npublic section\n\nlibrary_note «partially-applied ext lemmas»\n/--\nWhen possible, `ext` lemmas are stated without a full set of arguments. As an example, for bundled\nhoms `f`, `g`, and `of`, `f.comp of = g.comp of → f = g` is a better `ext` lemma than\n`(∀ x, f (of x) = g (of x)) → f = g`, as the former allows a second type-specific extensionality\nlemmas to be applied to `f.comp of = g.comp of`.\nIf the domain of `of` is `ℕ` or `ℤ` and `of` is a `RingHom`, such a lemma could then make the goal\n`f (of 1) = g (of 1)`.\n\nFor bundled morphisms, there is a `ext` lemma that always applies of the form\n`(∀ x, ⇑f x = ⇑g x) → f = g`. When adding type-specific `ext` lemmas like the one above, we want\nthese to be tried first. This happens automatically since the type-specific lemmas are inevitably\ndefined later.\n-/\n","size":1185,"sha256":"0a95c05b4752da973713a76463870227dde04ca4a2fe8998bcfe1c87f12edac4","revision":"de3a9cf33016bbb6d15880d7680643f7ca2d25ba","imports":["Mathlib.Init","Batteries.Util.LibraryNote"],"asset":"lean-mathlib/sources/Mathlib/Tactic/Ext.lean","url":"https://github.com/leanprover-community/mathlib4/blob/de3a9cf33016bbb6d15880d7680643f7ca2d25ba/Mathlib/Tactic/Ext.lean"},"Mathlib.Util.AssertNoSorry":{"module":"Mathlib.Util.AssertNoSorry","path":"Mathlib/Util/AssertNoSorry.lean","source":"/-\nCopyright (c) 2023 David Renshaw. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Renshaw\n-/\nmodule\n\npublic import Mathlib.Init\npublic meta import Lean.Util.CollectAxioms\npublic meta import Lean.Elab.Command\n\n/-!\n# Defines the `assert_no_sorry` command.\n\nThrows an error if the given identifier uses sorryAx.\n-/\n\npublic meta section\n\nopen Lean Meta Elab Command\n\n/-- Throws an error if the given identifier uses sorryAx. -/\nelab \"assert_no_sorry \" n:ident : command => do\n  let name ← liftCoreM <| Lean.Elab.realizeGlobalConstNoOverloadWithInfo n\n  let axioms ← Lean.collectAxioms name\n  if axioms.contains ``sorryAx\n  then throwError \"{n} contains sorry\"\n","size":719,"sha256":"aa9f7bebacafc688c894ef2171930e51ed19e0dfe722581848a2414d28900d4d","revision":"de3a9cf33016bbb6d15880d7680643f7ca2d25ba","imports":["Mathlib.Init","Lean.Util.CollectAxioms","Lean.Elab.Command"],"asset":"lean-mathlib/sources/Mathlib/Util/AssertNoSorry.lean","url":"https://github.com/leanprover-community/mathlib4/blob/de3a9cf33016bbb6d15880d7680643f7ca2d25ba/Mathlib/Util/AssertNoSorry.lean"},"Mathlib.Tactic.Positivity.Finset":{"module":"Mathlib.Tactic.Positivity.Finset","path":"Mathlib/Tactic/Positivity/Finset.lean","source":"/-\nCopyright (c) 2023 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nmodule\n\npublic import Mathlib.Algebra.Order.BigOperators.Group.Finset  -- shake: keep (Qq dependency)\npublic import Mathlib.Data.Finset.Density  -- shake: keep (Qq dependency)\npublic import Mathlib.Tactic.NormNum.Basic\n\n/-!\n# Positivity extensions for finsets\n\nThis file provides a few `positivity` extensions that cannot be in either the finset files (because\nthey don't know about ordered fields) or in `Tactic.Positivity.Basic` (because it doesn't want to\nknow about finiteness).\n-/\n\npublic meta section\n\nnamespace Mathlib.Meta.Positivity\n\nopen Qq Lean Meta Finset\n\n/-- Extension for `Finset.card`. `#s` is positive if `s` is nonempty.\n\nIt calls `Mathlib.Meta.proveFinsetNonempty` to attempt proving that the finset is nonempty. -/\n@[positivity Finset.card _]\nmeta def evalFinsetCard : PositivityExt where eval {u α} _ pα? e :=\n  match pα? with | none => pure .none | some _ => do\n  match u, α, e with\n  | 0, ~q(ℕ), ~q(Finset.card $s) =>\n    let some ps ← proveFinsetNonempty s | return .none\n    assertInstancesCommute\n    return .positive q(Finset.Nonempty.card_pos $ps)\n  | _ => throwError \"not Finset.card\"\n\n/-- Extension for `Fintype.card`. `Fintype.card α` is positive if `α` is nonempty. -/\n@[positivity Fintype.card _]\nmeta def evalFintypeCard : PositivityExt where eval {u α} _ pα? e :=\n  match pα? with | none => pure .none | some _ => do\n  match u, α, e with\n  | 0, ~q(ℕ), ~q(@Fintype.card $β $instβ) =>\n    let instβno ← synthInstanceQ q(Nonempty $β)\n    assumeInstancesCommute\n    return .positive q(@Fintype.card_pos $β $instβ $instβno)\n  | _ => throwError \"not Fintype.card\"\n\n/-- Extension for `Finset.dens`. `s.dens` is positive if `s` is nonempty.\n\nIt calls `Mathlib.Meta.proveFinsetNonempty` to attempt proving that the finset is nonempty. -/\n@[positivity Finset.dens _]\nmeta def evalFinsetDens : PositivityExt where eval {u 𝕜} _ pα? e :=\n  match pα? with | none => pure .none | some _ => do\n  match u, 𝕜, e with\n  | 0, ~q(ℚ≥0), ~q(@Finset.dens $α $instα $s) =>\n    let some ps ← proveFinsetNonempty s | return .none\n    assumeInstancesCommute\n    return .positive q(@Nonempty.dens_pos $α $instα $s $ps)\n  | _, _, _ => throwError \"not Finset.dens\"\n\nattribute [local instance] monadLiftOptionMetaM in\n/-- The `positivity` extension which proves that `∑ i ∈ s, f i` is nonnegative if `f` is, and\npositive if each `f i` is and `s` is nonempty.\n\nTODO: The following example does not work\n```\nexample (s : Finset ℕ) (f : ℕ → ℤ) (hf : ∀ n, 0 ≤ f n) : 0 ≤ s.sum f := by positivity\n```\nbecause `compareHyp` can't look for assumptions behind binders.\n-/\n@[positivity Finset.sum _ _]\nmeta def evalFinsetSum : PositivityExt where eval {u α} zα pα? e :=\n  match pα? with\n  | none => pure .none -- TODO: the case without PartialOrder\n  | some pα => do\n  match e with\n  | ~q(@Finset.sum $ι _ $instα $s $f) =>\n    let i : Q($ι) ← mkFreshExprMVarQ q($ι) .syntheticOpaque\n    have body : Q($α) := .betaRev f #[i]\n    let rbody ← core zα pα body\n    let p_pos : Option Q(0 < $e) ← do\n      let .positive pbody := rbody | pure none -- Fail if the body is not provably positive\n      let some ps ← proveFinsetNonempty s | pure none\n      let .some pα' ← trySynthInstanceQ q(IsOrderedCancelAddMonoid $α) | pure none\n      assertInstancesCommute\n      let pr : Q(∀ i, 0 < $f i) ← mkLambdaFVars #[i] pbody\n      pure <| some q(@sum_pos $ι $α $instα (@PartialOrder.toPreorder _ $pα) $pα' $f $s _\n        (fun i _ ↦ $pr i) $ps)\n    -- Try to show that the sum is positive\n    if let some p_pos := p_pos then\n      return .positive p_pos\n    -- Fall back to showing that the sum is nonnegative\n    else\n      let pbody ← rbody.toNonneg\n      let pr : Q(∀ i, 0 ≤ $f i) ← mkLambdaFVars #[i] pbody\n      let pα' ← synthInstanceQ q(AddLeftMono $α)\n      assertInstancesCommute\n      return .nonnegative q(@sum_nonneg $ι $α $instα (@PartialOrder.toPreorder _ $pα) $f $s $pα'\n        fun i _ ↦ $pr i)\n  | _ => throwError \"not Finset.sum\"\n\nvariable {α : Type*} {s : Finset α}\n\nexample : 0 ≤ #s := by positivity\nexample (hs : s.Nonempty) : 0 < #s := by positivity\n\nvariable [Fintype α]\n\nexample : 0 ≤ Fintype.card α := by positivity\nexample : 0 ≤ dens s := by positivity\nexample (hs : s.Nonempty) : 0 < dens s := by positivity\nexample (hs : s.Nonempty) : dens s ≠ 0 := by positivity\nexample [Nonempty α] : 0 < #(univ : Finset α) := by positivity\nexample [Nonempty α] : 0 < Fintype.card α := by positivity\nexample [Nonempty α] : 0 < dens (univ : Finset α) := by positivity\nexample [Nonempty α] : dens (univ : Finset α) ≠ 0 := by positivity\n\nexample {G : Type*} {A : Finset G} :\n    let f := fun _ : G ↦ 1; (∀ s, f s ^ 2 = 1) → 0 ≤ #A := by\n  intros\n  positivity -- Should succeed despite failing to prove `A` is nonempty.\n\nend Mathlib.Meta.Positivity\n","size":5034,"sha256":"6aab775f8ff057ee93263e7ea22a5f2ec0f6b871bef39bd8e8821005820ba386","revision":"de3a9cf33016bbb6d15880d7680643f7ca2d25ba","imports":["Mathlib.Algebra.Order.BigOperators.Group.Finset","Mathlib.Data.Finset.Density","Mathlib.Tactic.NormNum.Basic"],"asset":"lean-mathlib/sources/Mathlib/Tactic/Positivity/Finset.lean","url":"https://github.com/leanprover-community/mathlib4/blob/de3a9cf33016bbb6d15880d7680643f7ca2d25ba/Mathlib/Tactic/Positivity/Finset.lean","syntaxCheck":"frontend"},"Mathlib.Tactic.Positivity":{"module":"Mathlib.Tactic.Positivity","path":"Mathlib/Tactic/Positivity.lean","source":"module\n\npublic import Mathlib.Algebra.Order.Field.Basic\npublic import Mathlib.Tactic.Positivity.Basic\npublic import Mathlib.Tactic.Positivity.Finset\n\nset_option linter.style.header 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