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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 Management of the Lean search path (`LEAN_PATH`), which is a list of paths containing package roots: an import `A.B.C` resolves to `path/A/...
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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.subsemigroup.basic /-! # Submonoids: definition and...
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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 linear_algebra.free_module.basic import linear_algebra.matrix.to_lin import ring_theory.finiteness /-! # Finite and free modules We provide some instances...
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--import week_8.ideas.Z1 import week_8.ideas.distrub_mul_action_hom.Z1 -- do I need all of this? import week_8.ideas.Z1 /- # A crash course in H¹(G,M) We stick to the conventions that `G` is a group (or even a monoid, we never use inversion) and that `M` is a `G`-module, that is, an additive abelian group with a `G`...
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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, Johan Commelin, Mario Carneiro -/ import data.mv_polynomial.variables /-! # Multivariate polynomials over a ring Many results about polynomials hold when the coeffic...
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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, Sébastien Gouëzel, Rémy Degenne -/ import measure_theory.integral.set_to_l1 import analysis.normed_space.bounded_linear_maps import topology.sequences /...
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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 Tuples are lists of a fixed size. It is implemented as a subtype. -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.data.list.default import Mathlib.Lean3...
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/- Copyright (c) 2019 Jesse Michael Han. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author(s): Jesse Michael Han Tests for `finish using [...]` -/ import tactic.finish import algebra.order.ring.defs section list_rev open list variable {α : Type*} def append1 (a : α) : l...
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/- Copyright (c) 2020 Aaron Anderson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Aaron Anderson -/ import data.set.finite import order.atoms /-! # Atoms, Coatoms, Simple Lattices, and Finiteness This module contains some results on atoms and simple lattices in th...
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/- Copyright (c) 2014 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Module: int.basic Authors: Floris van Doorn, Jeremy Avigad The integers, with addition, multiplication, and subtraction. The representation of the integers is chosen to compute efficiently....
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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 Type of functions with finite support. Functions with finite support provide the basis for the following concrete instances: * ℕ →₀ α: Polynomials (where α is a ring...
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-- Lemmas import galois.data.list.nth_le namespace nat /-- Show that nat.to_digits never returns empty list. -/ theorem to_digits_ne_nil (base:ℕ) (x:ℕ) : nat.to_digits base x ≠ list.nil := begin cases x, case nat.zero : { simp [nat.to_digits], }, case nat.succ : x { simp [nat.to_digits], cases (nat.to_...
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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 -/ import Lean.Meta.KAbstract import Lean.Meta.Tactic.Util namespace Lean namespace Meta def generalize (mvarId : MVarId) (e : Expr) (x : Name) (failIfNotInT...
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import data.set open set section variable U: Type variables A B C: set U example : ∀ x, x ∈ A ∩ C → x ∈ A ∪ B := assume x, assume :x ∈ A ∩ C, have x ∈ A, from and.left this, show x ∈ A ∪ B, from or.inl this example : ∀ x, x ∈ (A ∪ B)ᶜ → x ∈ Aᶜ := assume x, assume :x ∈ (A ∪ B)ᶜ, h...
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/- Copyright 2020 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) 2020 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import category_theory.concrete_category.basic import category_theory.functor.reflects_isomorphisms /-! > THIS FILE IS SYNCHRONIZED WITH MATHLIB4. > Any changes to 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 Free abelian groups as abelianization of free groups. -/ import group_theory.free_group import group_theory.abelianization universes u v variables (α : Type u) ...
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/- Copyright (c) 2020 Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bhavik Mehta -/ import category_theory.limits.shapes.pullbacks import category_theory.limits.shapes.wide_pullbacks import category_theory.limits.shapes.binary_products import category_th...
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/- Copyright (c) 2019 Kevin Kappelmann. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kevin Kappelmann -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.algebra.continued_fractions.continuants_recurrence import Mathlib.algebra.continued_fra...
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/- ------------------------------------------------------------------------- -| | @project: riemann_hypothesis | | @file: riemann_zeta.lean | | @authors: Brandon H. Gomes, Alex Kontorovich | ...
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/- Copyright (c) 2022 Joël Riou. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Joël Riou -/ import algebraic_topology.dold_kan.homotopies import tactic.ring_exp /-! # Study of face maps for the Dold-Kan correspondence > THIS FILE IS SYNCHRONIZED WITH MATHLIB4. > A...
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example : (λ (u : Nat) => u + 0) = id :=by conv => lhs intro u change u
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/- Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro -/ import algebra.group_power.basic import algebra.order_functions import algebra.or...
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import game.sets.sets_level04 -- hide namespace xena -- hide open_locale classical -- hide variable X : Type --hide /- # Chapter 1 : Sets ## Level 5 -/ /- If `h : ∀ (x : X), x ∈ A → x ∈ B` then `h` is a function which takes a term `x` of type `X` as input, and also a proof that `x ∈ A`, and outputs a proof that ...
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/- Copyright (c) 2017 Gabriel Ebner. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Gabriel Ebner -/ import .clause .prover_state open tactic monad native namespace super private meta def try_subsume_core : list clause.literal → list clause.literal → tactic unit | []...
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import Init.Lean open Lean inductive Pattern | Inaccessible (e : Expr) | Var (fvarId : FVarId) | Ctor (fields : List Pattern) | Val (e : Expr) | ArrayLit (xs : List Pattern) structure AltLHS := (fvarIds : List FVarId) -- free variables used in the patterns (patterns : List Pattern) --...
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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, Patrick Massot, Casper Putz, Anne Baanen -/ import data.matrix.block import linear_algebra.matrix.finite_dimensional import linear_algebra.std_basis import ring_theory....
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open tactic lean lean.parser open interactive interactive.types expr local postfix *:9001 := many namespace tactic.interactive universes u v def congr_arg_f_equal {A : Sort u} {B : Sort v} {f f' : A → B} {x x' : A} : f = f' → x = x' → f x = f' x' := begin intros H, induction H, intros H, induction H, reflexivity ...
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open eq definition concat_pV_p {A : Type} {x y z : A} (p : x = z) (q : y = z) : (p ⬝ q⁻¹) ⬝ q = p := begin generalize p, apply (eq.rec_on q), intro p, apply (eq.rec_on p), apply idp end
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import algebra.precategory.basic open category context parameter {D₀ : Type} parameter (C : precategory D₀) parameter (D₂ : Π ⦃a b c d : D₀⦄ (f : hom a b) (g : hom c d) (h : hom a c) (i : hom b d), Type) attribute comp [reducible] definition comp₁_type [reducible] : Type := Π ⦃a b c₁ d₁ c₂ d₂ : D₀⦄ ...
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/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Leonardo de Moura, Floris van Doorn Partially ported from the standard library -/ import ..init universes u v w hott_theory namespace hott open decidable namespace bool -- ...
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-- Copyright (c) 2017 Scott Morrison. All rights reserved. -- Released under Apache 2.0 license as described in the file LICENSE. -- Authors: Tim Baumann, Stephen Morgan, Scott Morrison import category_theory.isomorphism import category_theory.functor_category open category_theory namespace category_theory.nat_iso ...
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/- Copyright (c) 2016 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl -/ import algebra.ordered_monoid /-! # Ordered groups This file develops the basics of ordered groups. ## Implementat...
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vm_eval #"\x41" vm_eval #"\x42" vm_eval #"\x43"
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import data.nat data.int open nat find_decl _ < succ _, +imp, -le find_decl _ < _ + _ find_decl _ = succ _ find_decl _ < succ _, +pos find_decl zero < succ _, pos set_option find_decl.expensive true -- Now, nat.zero will match 0 (i.e., (of_num num.zero)) find_decl zero < succ _, pos
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/- Copyright (c) 2020 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Mario Carneiro -/ import tactic.by_contra import data.set.image /-! # Well-founded relations > THIS FILE IS SYNCHRONIZED WITH MATHLIB4. > Any changes to this file requi...
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/- Copyright (c) 2019 Patrick Massot. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Patrick Massot -/ import tactic.monotonicity import order.basic open tactic interactive (parse) interactive (loc.ns) interactive.types (texpr location) lean.parser (tk) local p...
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@[simp] def iota : Nat → List Nat | 0 => [] | m@(n+1) => m :: iota n attribute [simp] List.iota
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import algebra.group.basic data.buffer universe u @[derive decidable_eq, derive has_reflect] inductive group_term : Type | of : ℕ → group_term | one : group_term | mul : group_term → group_term → group_term | inv : group_term → group_term namespace group_term instance : has_one group_term := ⟨group_term.one⟩ instan...
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/- Copyright (c) 2017 Simon Hudon. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Simon Hudon Instances for identity and composition functors -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.control.functor import Mathlib.algebra.group.basi...
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import data.nat open nat section variables (a b c d e : nat) theorem T (H1 : a = b) (H2 : b = c + 1) (H3 : c = d) (H4 : e = 1 + d) : a = e := by rewrite [H1, H2, H3, add.comm, -H4] end example (x y : ℕ) : (x + y) * (x + y) = x * x + y * x + x * y + y * y := by rewrite [*mul.left_distrib, *mul.right_distrib, -a...
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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 tactic.norm_cast /-! # Invertible elements This file defines a typeclass `invertibl...
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import data.real.basic theorem USAMO_Problem_1_1978 (a b c d e : real) : a+b+c+d+e = 8 ∧ a^2+b^2+c^2+d^2+e^2=16 → e ≤ 16/5 := sorry
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import system.io variable [io.interface] def main : io unit := do handle ← io.cmd {cmd := "echo", args := ["Hello World!"]}, io.put_str handle, return () #eval main
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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 tactic.lint import tactic.ext /-! # Sigma types This file proves basic results about sigma types. A sigma type is a dependent pair type. Like `α × β` but wh...
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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, Johannes Hölzl, Sander Dahmen, Scott Morrison -/ import algebra.module.big_operators import linear_algebra.dfinsupp import linear_algebra.free_module.basic import linea...
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/- Copyright (c) 2016 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl -/ import algebra.order.monoid.basic import algebra.order.monoid.cancel.defs /-! # Basic results on ordered cancellativ...
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meta def rwonce : tactic unit := do r ← tactic.to_expr `(FunctorComposition.associativity), a ← tactic.to_expr `(associator), tactic.rewrite_at_core reducible tt tt (occurrences.pos [2]) tt r a meta def rwonce_2 : tactic unit := do r ← tactic.to_expr `(switch_twice_is_the_identity), a ← tactic.to_expr `(switc...
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/- Copyright (c) 2020 Sebastian Ullrich. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sebastian Ullrich, Dany Fabian -/ import Lean.Meta.Transform import Lean.Elab.Deriving.Basic import Lean.Elab.Deriving.Util import Lean.Data.Json.FromToJson namespace Lean.Elab.Der...
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/- Copyright (c) 2021 Apurva Nakade. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Apurva Nakade -/ import set_theory.surreal.basic import ring_theory.localization /-! # Dyadic numbers Dyadic numbers are obtained by localizing ℤ away from 2. They are the initial obje...
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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.nat.interval /-! # Finite intervals of positive naturals This file proves that `ℕ+` is a `locally_finite_order` and calculates the cardinality of its interv...
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/- Copyright (c) 2020 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import topology.sheaves.sheaf import category_theory.sites.limits import category_theory.limits.functor_category /-! # Presheaves in `C` have limits and colimits whe...
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/- Copyright (c) 2017 Simon Hudon All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Simon Hudon, Mario Carneiro Tests for norm_num -/ import tactic.norm_num import data.complex.basic import data.nat.prime -- constant real : Type -- notation `ℝ` := real -- @[instance] c...
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def f [Add α] (a : α) : α := a + a def BV (n : Nat) := { a : Array Bool // a.size = n } def allZero (bv : BV n) : Prop := ∀ i, i < n → bv.val[i] = false def foo (b : BV n) (h : allZero b) : BV n := b def optbind (x : Option α) (f : α → Option β) : Option β := match x with | none => none | some a => f ...
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/- Copyright (c) 2014 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro Basic operations on the natural numbers. -/ import logic.basic algebra.ordered_ring data.option.basic universes u...
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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.uniform_space.completion import topology.metric_space.isometry /-! # The completion of a metric space Completion of uniform spaces are already...
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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 Some big operators for lists and finite sets. -/ import tactic.tauto data.list.defs data.finset data.nat.enat universes u v w variables {α : Type u} {β : Type v} {γ : ...
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/- Copyright (c) 2014 Parikshit Khanna. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro -/ import data.nat.basic /-! # Basic properties of lists -/ open function nat (hiding one_pos) ...
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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 Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.category_theory.limits.constructions.pullbacks import Mathlib.category_theory.limits.shapes.b...
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import init.lean.elaborator open Lean def main (xs : List String) : IO Unit := do initSearchPath "../../library:."; -- path ← IO.getEnv "LEAN_PATH"; -- IO.println path; contents ← IO.readTextFile xs.head; (env, messages) ← Elab.testFrontend contents xs.head; messages.toList.mfor $ fun msg => IO.println ...
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/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Leonardo de Moura -/ prelude import init.bool namespace pos_num protected definition mul (a b : pos_num) : pos_num := pos_num.rec_on a b (λ n r, bit0 r + b) (λ ...
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import mynat.definition import mynat.add import mynat.mul namespace mynat example : mynat → mynat := begin intro n, exact 3*n + 2, end end mynat
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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 tactic.split_ifs order.basic algebra.order algebra.ordered_group algebra.ring data.nat.cast universe u variable {α : Type u} -- TODO: this is necessary addit...
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/- Copyright (c) 2018 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Jeremy Avigad, Simon Hudon -/ import data.pfunctor.multivariate.basic universe u /-! # Multivariate quotients of polynomial functors. Basic definition of multivariate QPF. QPFs form a ...
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/- Copyright (c) 2014 Robert Lewis. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro -/ import algebra.ordered_ring import algebra.field set_option default_priority 100 -- see Note [default priority] set_option old_structu...
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/- Copyright (c) 2020. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jalex Stark, Yury Kudryashov -/ import tactic import data.real.basic /-! # Sum of the Reciprocals of the Triangular Numbers This file proves Theorem 42 from the [100 Theorems List](https://www.cs.r...
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import .unit import .graph meta def visualiser := tidy.rewrite_search.tracer.graph_tracer
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/- Copyright (c) 2016 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.list.basic import init.data.char.basic /- In the VM, strings are implemented using a dynamic array and UTF-8 encoding. TODO: we...
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/- Copyright (c) 2021 Yourong Zang. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yourong Zang, Yury Kudryashov -/ import topology.separation import topology.opens /-! # The Alexandroff Compactification We construct the Alexandroff compactification (the one-point co...
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/- Copyright (c) 2017 Johannes Hölzl. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Johannes Hölzl Extended non-negative reals -/ import topology.instances.nnreal data.real.ennreal noncomputable theory open classical set lattice filter metric open_locale classical ope...
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/- Copyright (c) 2019 Reid Barton. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel -/ import topology.constructions /-! # Neighborhoods and continuity relative to a subset This file defines relative versions * `nhds_within` of `nhds` * `co...
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/- Copyright (c) 2016 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Floris van Doorn The definition of pointed types. This file is here to avoid circularities in the import graph -/ prelude import init.trunc open eq equiv is_equiv is_trunc struct...
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-- constant f : nat → nat constant g : nat → nat → nat constant h : nat → nat → nat reserve infixl `+`:65 reserve infixr `&`:70 reserve infixl `-`:65 reserve prefix `-`:100 local infixl (name := «+») `+` := g local infixl (name := « - ») `-` := h local prefix (name := «- ») `-` := f local infixr (name := «&») `&` :...
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import ..tactic.all import ..cs.lang import ..cs.sym import ..cs.conc import ..op.sym import ..cs.util import .defs import ..cs.lib import ..op.simp import ..basic.basic import ..basic.sym import ..op.basic import .basic import .sym_factory namespace sym_factory open sym open op.sym open op.lang.op def ind_principle...
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/- Copyright (c) 2020 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Scott Morrison -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.category_theory.pi.basic import Mathlib.category_theory.limits.limits import Mathlib.PostPo...
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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.ResolveName import Lean.ScopedEnvExtension import Lean.Parser.Basic import Lean.Parser.StrInterpolation import Lean.KeyedDecl...
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import data.real.basic import combinatorics.simplicial_complex.basic import combinatorics.simplicial_complex.finite import combinatorics.simplicial_complex.sperner variables {m n : ℕ} local notation `E` := fin m → ℝ noncomputable def barycenter(L: list E): E := (list.foldl (λ x:E , λ y:E, x + y) 0 L) / L.length ...
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/- Copyright (c) 2015 Jacob Gross. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jacob Gross, Jeremy Avigad Open and closed sets, seperation axioms and generated topologies. -/ import data.set data.nat open algebra eq.ops set nat structure topology [class] (X : Type...
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/- Copyright (c) 2017 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Tim Baumann, Stephen Morgan, Scott Morrison Defines natural transformations between functors. Introduces notations `F ⟹ G` for the type of natural transformations between functors ...
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def main : IO Unit := do IO.println ((2 : Float).sin); IO.println ((2 : Float).cos); IO.println ((2 : Float).sqrt); IO.println ((2 : Float) ^ (200 : Float)); pure () #eval main
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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 vectors as list subtype -/ import logic data.list data.subtype open nat list subtype function definition vec [reducible] (A : Type) (n : nat) := {l : list A |...
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import data.nat open nat attribute nat.add [unfold 2] attribute nat.rec_on [unfold 2] infixl `;`:15 := tactic.and_then namespace tactic definition then_all (t1 t2 : tactic) : tactic := focus (t1 ; all_goals t2) end tactic tactic_infixl `;;`:15 := tactic.then_all open tactic example (a b c : nat) : (a + 0 = 0 +...
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/- Copyright (c) 2021 Kalle Kytölä. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kalle Kytölä -/ import tactic import measure_theory.measurable_space import measure_theory.integration import measure_theory.borel_space import topology.metric_space.basic import topolo...
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import standard structure has_mul [class] (A : Type) := mk :: (mul : A → A → A) check @has_mul.mul -- Π {A : Type} [c : has_mul A], A → A → A infixl `*` := has_mul.mul section variables (A : Type) (s : has_mul A) (a b : A) check a * b end
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import algebra.module import ring_theory.algebra import algebra.punit_instances variables (R : Type) [ring R] class modules := (space : Type) [acg : add_comm_group space] [mod : module R space] variable (M : modules R) instance modules_add_comm_group : add_comm_group M.space := modules.acg R instance modules_...
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constant f : nat → nat notation a `²` := f a variable a : nat #check a²
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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 In the standard library we cannot assume the univalence axiom. We say two types are equivalent if they are isomorphic. Two equivalent types have the same car...
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/- page 84 -/ import standard namespace hide -- BEGIN inductive Exists {A : Type} (P : A → Prop) : Prop := intro : ∀ (a : A), P a → Exists P definition exists.intro := @Exists.intro -- END end hide
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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 -/ import measure_theory.function.lp_space /-! # Integrable functions and `L¹` space In the first part of this file, the predicate `integrable` is defined and basic pro...
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/- Each fabstract contains a list of results. The following type lists the various kinds of results. You may read the values as follows: * `Proof p` -- the paper contains the proof `p` * `Construction c` -- the paper contains the construction `c` * `Conjecture C` -- the paper contains the conjecture `C` ...
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def f1 {α} [HasToString α] (a : α) : String := -- works ">> " ++ toString a -- Moving `{α} [HasToString α]` to a `variables` break the example variables {α} [HasToString α] def f2 (a : α) : String := ">> " ++ toString a class Dummy (α : Type) := (val : α) /- The following fails because `variables {α : Type _} [Dummy...
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import .lattice universes u variables {α β : Type u} section variables [preorder α] [preorder β] /-- A Galois connection is a pair of functions `l` and `u` satisfying `l a ≤ b ↔ a ≤ u b`. They are closely connected to adjoint functors in category theory. -/ def galois_connection (l : α → β) (u : β → α) := ∀ a ...
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/- Copyright (c) 2014 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Leonardo de Moura -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.logic.unique import Mathlib.order.boolean_algebra import Mathlib.PostPort ...
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/- Copyright (c) 2021 David Wärn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: David Wärn -/ import data.set.basic import combinatorics.quiver.basic /-! ## Wide subquivers A wide subquiver `H` of a quiver `H` consists of a subset of the edge set `a ⟶ b` for every p...
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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 data.dlist data.dlist.basic data.prod category.basic tactic.basic tactic.rcases tactic.generalize_proofs tactic.split_ifs meta.expr logic.basic tactic.ex...
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/- Copyright (c) 2020 Marc Huisinga. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Marc Huisinga, Wojciech Nawrocki -/ import Init.System.IO import Init.Data.ByteArray import Std.Data.RBMap import Lean.Elab.Import import Lean.Data.Lsp import Lean.Server.FileSource ...
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/- Copyright (c) 2018 Scott Morrison. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn -/ import category_theory.whiskering import category_theory.yoneda import category_theory.limits.cones import category_the...
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/- Copyright (c) 2014 Parikshit Khanna. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn -/ prelude import init.data.list.basic init.function init.meta init.data.nat.lemmas import init.meta.interactive i...
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import data.list import .string import .pretty_print meta def binder_info.brackets : binder_info → string × string | binder_info.default := ("(", ")") | binder_info.implicit := ("{", "}") | binder_info.inst_implicit := ("[", "]") | bi := ("?", "?" ++ repr bi) meta structure binder := (name : name) (type : expr) (bi...
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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: Daniel Selsam, Leonardo de Moura Type class instance synthesizer using tabled resolution. -/ import Lean.Meta.Basic import Lean.Meta.Instances import Lean.Meta.AbstractMVars im...