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/- Copyright (c) 2019 Seul Baek. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Seul Baek -/ /- Linear integer arithmetic formulas in pre-normalized form. -/ import tactic.omega.int.preterm namespace omega namespace int /-- Intermediate shadow syntax for LNA formul...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov -/ import tactic.tfae import order.liminf_limsup import data.set.intervals import topology.algebra.group import topology.constructions ...
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/- Copyright (c) 2019 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import order.filter.extr import topology.continuous_on /-! # Local extrema of functions on topological spaces ## Main definitions This file defines special versi...
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/- Basic lean examples -/ variables (α : Type) (p q : α → Prop) /- The keywords `assume` and `take` are just syntax sugar for λ. (show t, from p) is just syntax sugar for the type ascription (p : t) We can inspect the type of and.left by hovering over it. -/ example : (∀ x : α, p x ∧ q x) → ∀ y : α, p y := assume h...
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/- Copyright (c) 2016 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura -/ prelude import init.meta.name init.meta.options init.meta.format init.meta.rb_map import init.meta.level init.meta.expr init.meta.environment init.meta.attr...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl Binder elimination -/ import order tactic.converter.old_conv namespace old_conv open tactic monad meta instance : monad_fail old_conv := { fail := λ α s, (λr e, tacti...
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structure Payload := (key : Nat) (val : Nat) @[noinline] def get? (p : Payload) (k : Nat) : Option Nat := if p.key == k then p.val else none inductive T | a (i : Nat) | b (i : Nat) | c (i : Nat) | d (i : Nat) @[noinline] def foo (p : Payload) : Option T := (do i ← get? p 1; pure (T.a i)) <|> (do i ← get? p 2...
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/- Copyright (c) 2021 Andrew Yang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Andrew Yang, Yury G. Kudryashov -/ import topology.continuous_on import data.setoid.basic import tactic.tfae /-! # Inseparable points in a topological space In this file we define * `s...
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open nat prod open decidable constant modulo' (x : ℕ) (y : ℕ) : ℕ infixl `mod`:70 := modulo' constant gcd_aux : ℕ × ℕ → ℕ noncomputable definition gcd' (x y : ℕ) : ℕ := gcd_aux (x, y) theorem gcd_def (x y : ℕ) : gcd' x y = @ite nat (y = 0) (nat.decidable_eq (snd (x, y)) 0) x (gcd' y (x mod y)) := sorry theorem gcd...
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import init.data.nat.basic import init.data.fin.basic import data.vector import .Prelude open Staged open nat open fin open vector section LeftSpindle structure LeftSpindle (A : Type) : Type := (linv : (A → (A → A))) (leftDistributive : (∀ {x y z : A} , (linv x (linv y z)) = (linv (linv x y) (linv ...
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/- Comment -/ -- comment -- declaring objects constants m n : nat constant f : nat → nat constant p : prod nat nat #check (m, n) #check p.1 -- types are objects #check nat constants α β : Type constant F : Type → Type #check α #check F α -- of what type is the type Type? #check Type #check Type 1 #check Type 2 -...
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/- Copyright (c) 2020 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura Extra notation that depends on Init/Meta -/ prelude import Init.Meta import Init.Data.Array.Subarray import Init.Data.ToString namespace Lean syntax "Macro.t...
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/- Copyright (c) 2017 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro -/ import logic.basic data.bool init.data.option.instances tactic.interactive namespace option variables {α : Type*} {β : Type*} instance has_mem : has_mem α (o...
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import ..u_semiring import .cosette_lemmas open nat open list section cosette_tactics def list.remove_first_of_each {α : Type} [decidable_eq α] : list α → list α → list α | xs [] := xs | [] ys := [] | (x::xs) (y::ys) := if x = y then list.remove_first_of_each xs ys els...
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/- Copyright (c) 2021 Oliver Nash. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Oliver Nash -/ import algebra.lie.submodule import algebra.lie.of_associative import linear_algebra.isomorphisms /-! # Quotients of Lie algebras and Lie modules > THIS FILE IS SYNCHRONI...
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/- Copyright (c) 2019 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import data.dfinsupp /-! # Direct sum This file defines the direct sum of abelian groups, indexed by a discrete type. ## Notation `⨁ i, β i` is the n-ary direct sum `direc...
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/- Copyright (c) 2021 Patrick Massot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Massot -/ import algebra.order.with_zero import topology.algebra.order.basic /-! # The topology on linearly ordered commutative groups with zero Let `Γ₀` be a linearly ordere...
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/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir -/ import algebra.geom_sum import data.nat.choose import data.complex.basic /-! # Exponential, trigonometric and hyperbolic trigonometric functions...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import topology.instances.nnreal import topology.algebra.order.monotone_continuity import analysis.normed.group.basic /-! # Extended non-negative reals -/ noncomputa...
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/- Copyright (c) 2019 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura -/ import Lean.Parser.Command import Lean.ResolveName import Lean.Meta.Reduce import Lean.Elab.Log import Lean.Elab.Term import Lean.Elab.Binders import Lean.E...
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import data.finset universes u open finset variables {α : Type u} [decidable_eq α] [fintype α] /- A matroid M is an ordered pair `(E, ℐ)` consisting of a finite set `E` and a collection `ℐ` of subsets of `E` having the following three properties: (I1) `∅ ∈ ℐ`. (I2) If `I ∈ ℐ` and `I' ⊆ I`, then `I' ∈ ℐ`. (I3...
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/- Copyright (c) 2018 Patrick Massot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Massot, Mario Carneiro, Johan Commelin Various multiplicative and additive structures. -/ import algebra.group.basic variables {α : Type*} {β : Type*} [group α] [group β] /-...
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import Lean.Server #eval Lean.Server.Test.runWithInputFile "./edits_client.log" none -- The builtin search path seems to be fine
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/- Copyright (c) 2018 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Johannes Hölzl -/ import data.finset.basic import data.multiset.pi /-! # The cartesian product of finsets -/ namespace finset open multiset /-! ### pi -/ section pi variables {α : Ty...
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/- Copyright (c) 2015 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Leonardo de Moura Type class for encodable types. Note that every encodable type is countable. -/ import data.fintype data.list data.list.sort data.sum data.nat.div data.countab...
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/- Copyright (c) 2019 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import algebra.category.Group.preadditive import group_theory.quotient_group import category_theory.limits.concrete_category import category_theory.limits.shapes.kern...
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def Do_Twice {α : Type} (f : α → α) (x : α) : α := f (f x)
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/- Copyright (c) 2019 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import order.filter.extr topology.continuous_on /-! # Local extrema of functions on topological spaces ## Main definitions This file defines special versions of...
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/- Copyright (c) 2016 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura -/ prelude import init.meta.interactive universes u v def state (σ α : Type u) : Type u := σ → α × σ section variables {σ α β : Type u} @[inline] def state_...
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/- Copyright (c) 2019 Gabriel Ebner. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Gabriel Ebner, Sébastien Gouëzel -/ import analysis.calculus.fderiv /-! # One-dimensional derivatives This file defines the derivative of a function `f : 𝕜 → F` where `𝕜` is a norm...
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import Lean open Lean initialize blaAttr : TagAttribute ← registerTagAttribute `bla "simple user defined attribute"
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import data.list open nat list context parameter {A : Type} parameter (p : A → Prop) parameter [H : decidable_pred p] include H definition filter : list A → list A | filter nil := nil | filter (a :: l) := if p a then a :: filter l else filter l theorem filter_nil : filter nil = nil := rfl t...
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import ..u_semiring import ..sql import ..tactics
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import data.nat.choose data.nat.prime data.fintype.basic algebra.big_operators.basic import tactic.ring /- --------------------------------------------------------------- -/ -- Section 3.1.2 #check Prop #check Type #check Type 5 def my_matrix (R : Type) (n m : ℕ) : Type := (fin n) → (fin m) → R def my_matrix.mul ...
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/- Copyright (c) 2019 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes ## Lagrange's four square theorem The main result in this file is `sum_four_squares`, a proof that every natural number is the sum of four square numbers. # Implementatio...
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/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth -/ import measure_theory.function.l1_space import measure_theory.function.lp_order /-! # Density of simple functions Show that each Bor...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl -/ import algebra.big_operators.basic import data.finset.pi import data.finset.powerset /-! # Results about big operators with values in a (semi)ring We prove results...
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/- Copyright (c) 2019 Zhouhang Zhou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Zhouhang Zhou The pointwise operations on filters have nice properties, such as • map m (f₁ * f₂) = map m f₁ * map m f₂ • 𝓝 x * 𝓝 y = 𝓝 (x * y) -/ import algebra.pointwise imp...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro -/ import topology.instances.int /-! # Topology on the natural numbers > THIS FILE IS SYNCHRONIZED WITH MATHLIB4. > Any changes to this file require a c...
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/- Copyright (c) 2019 Simon Hudon. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Simon Hudon -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.tactic.monotonicity.basic import Mathlib.control.traversable.default import Mathlib.control.traver...
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/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Kenny Lau, Joey van Langen, Casper Putz Characteristic of semirings. -/ import data.padics.padic_norm data.nat.choose data.fintype import data.zmod.basic algebra.module universes u v /--...
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/- Copyright (c) 2019 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura -/ import Lean.ResolveName import Lean.Util.Sorry import Lean.Structure import Lean.Meta.ExprDefEq import Lean.Meta.AppBuilder import Lean.Meta.SynthInstance i...
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/- Copyright (c) 2018 Mario Carneiro. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Mario Carneiro, Chris Hughes -/ import algebra.gcd_monoid.basic import algebra.euclidean_domain.basic import ring_theory.ideal.basic import ring_theory.principal_ideal_domain /-! # Le...
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/- Copyright (c) 2019 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Leonardo de Moura -/ prelude import Init.Data.Nat.Linear import Init.Data.List.Basic import Init.Util universe u namespace List /-! The following functions can't be defined at ...
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/- Copyright (c) 2021 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import category_theory.preadditive.default import category_theory.preadditive.single_obj import category_theory.preadditive.additive_functor import category_theory.li...
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/- Copyright (c) 2018 Simon Hudon. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Simon Hudon -/ import control.functor /-! # Traversable type class Type classes for traversing collections. The concepts and laws are taken from <http://hackage.haskell.org/package/base...
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/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import topology.metric_space.baire import analysis.normed_space.operator_norm import analysis.normed_space.affine_isometry /-! # Banach open mapping theorem T...
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/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau, Joey van Langen, Casper Putz -/ import algebra.hom.iterate import data.int.modeq import data.nat.choose.dvd import data.nat.choose.sum import group_theory.order_of_element impor...
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/- Copyright © 2020 Nicolò Cavalleri. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Nicolò Cavalleri, Yury Kudryashov -/ import geometry.manifold.cont_mdiff_map /-! # Diffeomorphisms This file implements diffeomorphisms. ## Definitions * `diffeomorph I I' M M' n`:...
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inductive point (A B : Type*) | mk : Π (x : A) (y : B), point inductive [class] color | red | green | blue constant {u} foo (A : Type u) [H : decidable_eq A] : Type u constants a : num section universe variable l variable A : Type.{l} variable Ha : decidable_eq A include Ha variable E : Type 1 include E...
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/- Copyright (c) 2019 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Sebastian Ullrich -/ prelude import Init.Lean.Elab.Import import Init.Lean.Elab.Command namespace Lean namespace Elab namespace Frontend structure Context :...
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/- Copyright (c) 2018 Simon Hudon. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Simon Hudon, Patrick Massot -/ import algebra.group.pi import algebra.ordered_group import tactic.pi_instances /-! # Pi instances for ordered groups and monoids This file defines instanc...
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/- Copyright (c) 2015 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Mario Carneiro In the standard library we cannot assume the univalence axiom. We say two types are equivalent if they are isomorphic. Two equivalent types h...
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/- Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies, Bhavik Mehta -/ import algebra.hom.freiman import analysis.asymptotics.asymptotics import analysis.convex.strict_convex_space /-! # Salem-Spencer sets and Ro...
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import data.multiset import data.list_extra namespace multiset open multiset variables {α : Type*} def all_prop (p : α → Prop) (s : multiset α) : Prop := quot.lift_on s (list.all_prop p) (λ l₁ l₂ e, by {change list.perm l₁ l₂ at e, apply propext, apply @list.perm_induction_on α (λ l₁ l₂, list.all_prop p l₁ ↔ ...
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/- Copyright (c) 2020 Markus Himmel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Markus Himmel -/ import category_theory.limits.shapes.constructions.pullbacks import category_theory.limits.shapes.regular_mono import category_theory.limits.shapes.biproducts import c...
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/- Copyright 2021 Google LLC Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software di...
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/- Copyright (c) 2021 Riccardo Brasca. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Riccardo Brasca -/ import data.set.finite import group_theory.submonoid.operations import group_theory.subgroup /-! # Finitely generated monoids and groups We define finitely gener...
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/- Copyright (c) 2019 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Leonardo de Moura -/ prelude import Init.Data.Array.Basic import Init.Data.Array.Subarray import Init.Data.UInt import Init.Data.Option.Basic universe u structure ByteArray wher...
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/- Copyright (c) 2021 Damiano Testa. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Damiano Testa -/ import algebra.group_power.basic import algebra.ring.opposite import group_theory.group_action.opposite import group_theory.group_action.prod /-! # Introduce `smul_wit...
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/- Copyright (c) 2018 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard, Amelia Livingston, Yury Kudryashov -/ import group_theory.submonoid.operations import algebra.big_operators.ba...
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/- Copyright (c) 2020 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin -/ import ring_theory.witt_vector.is_poly /-! ## Multiplication by `n` in the ring of Witt vectors In this file we show that multiplication by `n` in the ring of Witt...
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import data.set import data.nat.sqrt open set namespace relation_2102_ns /- We're now going to formally explain what a "section" is for in Lean. A section allows you to specify in one place a set of parameters that are then assumed to be arguments to every other definition in the section. Here's a very short littl...
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/- Copyright (c) 2021 Heather Macbeth. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Heather Macbeth -/ import field_theory.finite.basic /-! # IMO 2005 Q4 Problem: Determine all positive integers relatively prime to all the terms of the infinite sequence `a n = 2 ^ ...
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/- The issue this is checking is whether vm_overrides are preserved in oleans. It will segfault if not. -/ #check tactic.save_info_with_widgets run_cmd tactic.save_info_with_widgets ⟨1,2⟩ *> tactic.trace "hello" run_cmd tactic.save_info ⟨1,2⟩ *> tactic.trace "hello" -- sometimes segfaults because inliner using a b...
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import topology.algebra.ordered import o_minimal.order namespace omin open o_minimal universe u variables (R : Type u) [decidable_linear_order R] (S : struc R) variables {X : Type*} [has_coordinates R X] variables {Y : Type*} [has_coordinates R Y] variables {Z : Type*} [has_coordinates R Z] variables {W : Type*} [...
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class OPClass (α : outParam Type) (β : Type) : Type := (u : Unit := ()) instance op₁ : OPClass Nat Nat := {} new_frontend set_option pp.all true #synth OPClass Nat Nat
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/- Copyright (c) 2017 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura -/ namespace Std universes u v w w' inductive Rbcolor where | red | black inductive RBNode (α : Type u) (β : α → Type v) where | leaf ...
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/- Copyright (c) 2017 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Sebastian Ullrich -/ inductive Color | red | black inductive Tree where | leaf | node : Color → Tree → Nat → Bool → Tree → Tree deriving Inhabited ...
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/- Copyright (c) 2019 Seul Baek. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Seul Baek -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.tactic.omega.coeffs import Mathlib.PostPort namespace Mathlib /- Normalized linear integer arithmeti...
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import data.real.basic import data.vector open real variable X : Type -- hide /- # Work in progress. -/ -- begin hide #check finset #check finset.card_range #check list --Do we want a list? A vector? --either way, not sure how to go about this yet --def partition (a b : ℝ) (A: set.Icc a b) (n : ℕ) := list ℝ -- ...
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import order.filter.basic import for_mathlib.data.set.basic local infixr ` ×ᶠ `:51 := filter.prod section open filter set function lemma filter.comap_pure {α : Type*} {β : Type*} {f : α → β} (h : injective f) (a : α) : pure a = comap f (pure $ f a) := by rw [← filter.map_pure, comap_map h] variables {α : Type*} ...
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/- Copyright (c) 2018 Patrick Massot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Massot, Johannes Hölzl Completion of topological groups: -/ import topology.uniform_space.completion topology.algebra.uniform_group noncomputable theory section group open un...
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import category_theory.limits.shapes.biproducts /-! Let's show that every preadditive category embeds into a preadditive category with biproducts, and identify a good universal property. -/ universes v u variables (C : Type u) structure family := (ι : Type v) [fintype : fintype ι] [decidable_eq : decidable_eq ι] (v...
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/- Copyright (c) 2020 Thomas Browning and Patrick Lutz. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Thomas Browning and Patrick Lutz -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.field_theory.intermediate_field import Mathlib.field_th...
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abbrev M := ReaderT String $ StateT Nat $ ReaderT Bool $ IO def f : M Nat := do s ← read; IO.println s; b ← readThe Bool; IO.println b; s ← get; pure s #eval (f "hello").run' 10 true def g : M Nat := let a : M Nat := adaptTheReader Bool not f; adaptReader (fun s => s ++ " world") a #eval (g "hello").run' 10 true
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import data.list.dict import data.multiset import data.pfun namespace multiset universes u v variables {α : Type u} {β : α → Type v} open function list def nodupkeys (m : multiset (sigma β)) : Prop := quotient.lift_on m nodupkeys (λ _ _, propext ∘ perm_nodupkeys) theorem nodupkeys_coe {l : list (sigma β)} : nodup...
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/- Copyright (c) 2021 Heather Macbeth. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Heather Macbeth -/ import analysis.complex.circle import analysis.inner_product_space.l2_space import measure_theory.function.continuous_map_dense import measure_theory.function.l2_sp...
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/- Copyright (c) 2020 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura, Sebastian Ullrich -/ import Lean.Modifiers import Lean.Elab.Attributes import Lean.Elab.Exception import Lean.Elab.DeclUtil namespace Lean.Elab def checkNot...
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/- Copyright (c) 2019 Amelia Livingston. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Amelia Livingston, Jireh Loreaux -/ import algebra.ring.basic /-! # Homomorphisms of semirings and rings This file defines bundled homomorphisms of (non-unital) semirings and ring...
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/- Copyright (c) 2022 Moritz Doll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Moritz Doll -/ import analysis.locally_convex.basic import order.closure /-! # Balanced Core and Balanced Hull ## Main definitions * `balanced_core`: The largest balanced subset of a s...
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new_frontend def g : List Nat → List Nat → Nat | [], y::ys => y | [], ys => 0 | x1::x2::xs, ys => g xs ys | x::xs, y::ys => g xs ys + y | x::xs, [] => g xs [] universes u v inductive Imf {α β} (f : α → β) : β → Type _ | mk : (a : α) → Imf f (f a) def h {α β} {f : α → β} : {b : β} ...
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/- Copyright (c) 2020 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Anne Baanen A typeclass for the two-sided multiplicative inverse. -/ import algebra.char_zero import algebra.char_p /-! # Invertible elements This file defines a typeclass `invertible ...
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import pca namespace pca universe variable u variables {α : Type u} variables [pca α] /- Minimum submodel of pca containing A -/ inductive submodel (A : set α) : set α | rel {a} : a ∈ A → submodel a | k : submodel k | s : submodel s | mul {a b c} : (↓a * ↓b) = ↓c → submodel a → submodel b → submodel c notation `ℳ` :...
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-- DO NOT EDIT, automatically generated file, generator scripts/gen_constants_cpp.py import smt system.io open tactic meta def script_check_id (n : name) : tactic unit := do env ← get_env, (env^.get n >> return ()) <|> (guard $ env^.is_namespace n) <|> (attribute.get_instances n >> return ()) <|> fail ("identifier '" +...
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import Mathlib.Tactic.Basic example : (α : Type) × List α := by constructor · exact [0,1] -- example : (α : Type) × List α := by -- fsplit -- - exact ℕ -- - exact [0,1]
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/- Copyright (c) 2018 Kevin Buzzard and Patrick Massot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kevin Buzzard, Patrick Massot. This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl. -/ import group_theory.coset universes u v names...
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/- BinaryTools: Utilities for displaying and manipulating Binary data. -/ /- Simplification rules for ensuring type safety of Blake3Hash -/ @[simp] theorem ByteArray.size_empty : ByteArray.empty.size = 0 := rfl @[simp] theorem ByteArray.size_push (B : ByteArray) (a : UInt8) : (B.push a).size = B.size + 1 := by { case...
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print "parametric meta definition" meta definition f {A : Type} : nat → A → A → A | n a b := if n / 2 = 0 then a else f (n / 2) b a vm_eval if f 10 1 2 = 2 then "OK" else "FAILED" namespace foo print "parametric meta definition inside namespace" meta definition bla {A : Type} : nat → A → A → A | n a b := if n / 2 =...
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/- Copyright (c) 2018 Robert Y. Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Y. Lewis -/ import algebra.group_with_zero.power import algebra.ring.equiv import tactic.linarith /-! # Integer power operation on fields and division rings This file collect...
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/- Copyright (c) 2019 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Bhavik Mehta -/ import category_theory.monad.basic import category_theory.monad.kleisli import category_theory.category.Kleisli import category_theory.types /-! # Con...
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theorem ex1 (h₁ : ¬ p) (h₂ : q → p) : ¬ q := by intro h apply h₁ apply h₂ apply h
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/- Copyright (c) 2021 Yaël Dillies. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yaël Dillies -/ import data.sigma.lex import order.bounded_order /-! # Orders on a sigma type > THIS FILE IS SYNCHRONIZED WITH MATHLIB4. > Any changes to this file require a correspond...
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/- Copyright (c) 2019 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Simon Hudon The writer monad transformer for passing immutable state. -/ import algebra.group.defs import logic.equiv.basic universes u v w u₀ u₁ v₀ v₁ structure writer_t (ω ...
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import data.buffer.parser import init.data.unsigned import tools.json open parser def Ws : parser unit := decorate_error "<whitespace>" $ many' $ one_of' " \t\x0d\n".to_list def tok (s : string) := str s >> Ws def StringChar : parser char := sat (λc, c ≠ '\"' ∧ c ≠ '\\' ∧ c.val > 0x1f) <|> (do str "\\", (...
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constants A B : Type constant F : Type → Type constant G : Type → Type → Type check A -- Type check F A -- Type check F nat -- Type check G A -- Type → Type check G A B -- Type check G A nat -- Type
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/- Copyright (c) 2017 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sebastian Ullrich The identity Monad. -/ prelude import Init.Core universe u def Id (type : Type u) : Type u := type namespace Id @[inline] protected def pure {α : Type u} ...
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/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import analysis.special_functions.trigonometric.complex /-! # The `arctan` function. Inequalities, de...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johannes Hölzl, Mario Carneiro Finite sets. -/ import data.set.lattice data.nat.basic logic.function data.fintype open set lattice function universes u v w variables {α : Typ...