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/- Copyright (c) 2022 Michael Stoll. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Michael Stoll -/ import number_theory.legendre_symbol.zmod_char import field_theory.finite.basic /-! # Quadratic characters of finite fields This file defines the quadratic character ...
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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, Chris Hughes -/ import algebra.associated import algebra.big_operators.basic import ring_theory.valuation.basic /-! # Multiplicity of a divisor For a commutative mo...
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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 algebra.punit_instances import linear_algebra.finsupp import ring_theory.nilpotent import ring_theory.localization.away import ring_theory.ideal.prod import ri...
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/- Copyright (c) 2019 Neil Strickland. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Neil Strickland This contains some additional lemmas for working with heterogenous equality. It is far from clear that the approach taken here (or in places where these results ar...
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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 Converter monad for building simplifiers. -/ prelude import init.meta.tactic init.meta.simp_tactic init.meta.interactive import init.meta.congr_lemma init.met...
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/- page 19 -/ import standard constants A B C : Type constants (g : B → C) (f : A → B) -- BEGIN definition g_comp_f (x : A) : C := g (f x) print g_comp_f -- END
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import data.matrix.notation variables {α : Type} [semiring α] namespace matrix open_locale matrix example {a a' b b' c c' d d' : α} : ![![a, b], ![c, d]] + ![![a', b'], ![c', d']] = ![![a + a', b + b'], ![c + c', d + d']] := by simp example {a a' b b' c c' d d' : α} : ![![a, b], ![c, d]] ⬝ ![![a', b'], ![c', d...
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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.local_homeomorph /-! # Manifolds A manifold is a topological space M locally modelled on a model space H, i.e., the manifold is covered by op...
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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.Util.ForEachExpr import Lean.Elab.Term import Lean.Elab.Tactic.Basic namespace Lean.Elab.Term open Tactic (TacticM evalTacti...
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namespace dictionary meta def tactic.dec_trivial := `[exact dec_trivial] /- Definitions -/ section defs variables (κ α : Type*) def dict := κ → α instance [inhabited α] : inhabited (dict κ α) := ⟨λ _, default α⟩ end defs /- Operations and lemmas -/ section ops variables {κ α : Type*} [decidable_eq κ] def dict.up...
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import standard bool using bool namespace set definition set (T : Type) := T → bool definition mem {T : Type} (a : T) (s : set T) := s a = '1 infix `∈`:50 := mem section parameter {T : Type} definition empty : set T := λx, '0 notation `∅` := empty theorem mem_empty (x : T) : ¬ (x ∈ ∅) := not_intro (λH : ...
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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.tactic namespace tactic meta_definition set_bool_option (n : name) (v : bool) : tactic unit := do s ← read, write $ tactic_stat...
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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 -/ import algebra.direct_sum.basic import linear_algebra.dfinsupp /-! # Direct sum of modules over commutative rings, indexed by a discrete type. This file provides constructors...
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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, Jeremy Avigad, Yury Kudryashov, Patrick Massot -/ import order.filter.bases import data.finset.preimage /-! # `at_top` and `at_bot` filters on preorded sets, monoids a...
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/- Copyright (c) 2021 Eric Wieser. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Wieser -/ import linear_algebra.finsupp import algebra.monoid_algebra.support import algebra.direct_sum.internal import ring_theory.graded_algebra.basic /-! # Internal grading of an...
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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, Jeremy Avigad, Haitao Zhang General operations on functions. -/ prelude import init.data.prod init.funext init.logic universes u₁ u₂ u₃ u₄ namespace function...
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/- Copyright (c) 2020 Zhangir Azerbayev. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Wieser, Zhangir Azerbayev -/ import linear_algebra.multilinear import linear_algebra.linear_independent import group_theory.perm.sign import group_theory.perm.subgroup import ...
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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 Definitions and properties of div and mod. Much of the development follows Isabelle's library. -/ import data.nat.sub open eq.ops well_founded decidabl...
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/- Copyright (c) 2020 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anne Baanen -/ import linear_algebra.matrix.bilinear_form import linear_algebra.matrix.charpoly.minpoly import linear_algebra.determinant import linear_algebra.finite_dimensional import ...
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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 analysis.convex.basic import analysis.normed_space.bounded_linear_maps import analysis.specific_limits /-! # Tangent cone In this file, we define two p...
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/- Copyright (c) 2020 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Yury Kudryashov -/ import Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.algebra.invertible import Mathlib.linear_algebra.affine_space.affine_equiv import Mathlib....
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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, Patrick Stevens -/ import data.nat.choose.basic import data.nat.prime /-! # Divisibility properties of binomial coefficients -/ namespace nat open_locale nat namespace ...
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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 -/ import group_theory.perm.sign import group_theory.order_of_element namespace equiv.perm open equiv function finset variables {α : Type*} {β : Type*} [decidable_eq α] de...
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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: Jeremy Avigad Examples to test finish. (The designations "clarify," "safe," "iauto," etc. are from a previous tableau prover.) -/ import tactic.finish open auto section variab...
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/- Copyright (c) 2022 Eric Wieser. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Eric Wieser -/ import data.nat.log import algebra.order.floor import algebra.field_power /-! # Integer logarithms in a field with respect to a natural base This file defines two `ℤ`-val...
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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.int.gcd import tactic.abel /-! # Congruences modulo a natural number > THIS FILE IS SYNCHRONIZED WITH MATHLIB4. > Any changes to this file require a cor...
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import elliptic_curve_rationals import group_theory.subgroup import group_theory.finiteness import group_theory.quotient_group import data.fintype.card import data.set.finite import init.data.nat.lemmas import data.nat.prime import data.zmod.basic import data.complex.basic import analysis.special_functions.pow import o...
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/- Copyright (c) 2019 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Chris Hughes -/ import data.zsqrtd.basic import data.complex.basic import ring_theory.principal_ideal_domain import number_theory.quadratic_reciprocity /-! # Gaussian integers The Gaussi...
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/- Copyright (c) 2019 Johan Commelin. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Johan Commelin, Kenny Lau -/ import data.mv_polynomial import linear_algebra.std_basis import ring_theory.ideal.operations import ring_theory.multiplicity import ring_theory.algebra_to...
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/- Copyright (c) 2020 Joseph Myers. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Joseph Myers. -/ import data.finset.sort import linear_algebra.affine_space.combination import linear_algebra.basis noncomputable theory open_locale big_operators open_locale classical ...
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inductive Syntax : Type | I | II | III | IV | V inductive Semantics : Type | one | two | three | four | five #reduce one -- Visible only in Semantics namespace #reduce Semantics.one -- Qualified access to that namespace open Semantics -- Open namespaces open Syntax #reduce one -- Now de...
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/- Copyright (c) 2015 Nathaniel Thomas. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Nathaniel Thomas, Jeremy Avigad Modules and vector spaces over a ring. -/ import algebra.field data --data.matrix open eq.ops set function fin definition vector [reducible] (A : Ty...
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import Lean set_option linter.suspiciousUnexpanderPatterns true @[app_unexpander List.nil] def unexpandListNilBad : Lean.PrettyPrinter.Unexpander | `(List.nil) => `([]) | _ => throw () /--hey-/ @[app_unexpander List.cons] private def unexpandListConsBad : Lean.PrettyPrinter.Unexpander | `(List.cons $x []...
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import tactic.linarith example (e b c a v0 v1 : ℚ) (h1 : v0 = 5*a) (h2 : v1 = 3*b) (h3 : v0 + v1 + c = 10) : v0 + 5 + (v1 - 3) + (c - 2) = 10 := by linarith example (ε : ℚ) (h1 : ε > 0) : ε / 2 + ε / 3 + ε / 7 < ε := by linarith example (x y z : ℚ) (h1 : 2*x < 3*y) (h2 : -4*x + z/2 < 0) (h3 : 12*y - z < ...
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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.group.instances import algebra.order.monoid.type_tags /-! # Ordered group structures on `multip...
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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.finite_products import category_theory.limits.shapes.kernels import category_theory.limits.shapes.normal_mono import category_theor...
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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.expr universe variables u /- Quoted expressions. They can be converted into expressions by using a tactic. -/ meta_constant pexpr ...
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/- Copyright (c) 2021 Bhavik Mehta. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Bhavik Mehta -/ import category_theory.arrow import category_theory.limits.constructions.epi_mono import category_theory.limits.creates import category_theory.limits.preserves.finite imp...
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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 Std.ShareCommon import Lean.Parser.Command import Lean.Util.CollectLevelParams import Lean.Util.FoldConsts import Lean.Elab.Collec...
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/- Copyright (c) 2022 Rémi Bottinelli. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Rémi Bottinelli -/ import category_theory.category.basic import category_theory.functor.basic import category_theory.groupoid import tactic.nth_rewrite import category_theory.path_cat...
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/- Copyright (c) 2022 Thomas Browning. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Thomas Browning -/ import data.polynomial.unit_trinomial import ring_theory.polynomial.gauss_lemma import tactic.linear_combination /-! # Irreducibility of Selmer Polynomials > THI...
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def longArray (n : Nat := 50000) (xs : Array Char := #[]) : Array Char := match n with | 0 => xs | n+1 => longArray n (xs.push 'a') def OverflowIte {m : Type -> Type} [inst1: Monad m] (xs: Array Char) : StateT Nat m Nat := xs.foldlM (fun (len : Nat) (s : Char) => if s = 'z' then panic "z" else return len + 1...
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/- Copyright (c) 2018 Koundinya Vajjha. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Koundinya Vajjha A meta def called `#depends` which gives the names of all the theorems (the statement of) a given definition/theorem depends on. -/ import ..basic open tactic exp...
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import Lean open Lean Meta #eval show MetaM Unit from do let e ← withLetDecl `y (mkConst ``Nat) (mkConst ``Nat.zero) fun y => do let m ← mkFreshExprMVar (mkConst ``Nat) assignExprMVar m.mvarId! y let e ← mkApp2 (mkConst ``Nat.add) m y -- goal: construct λ y, e dbg_trace (← ppExpr (← mkLambdaFVars...
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/- page 36 -/ import standard constants p q : Prop -- BEGIN theorem t1 : p → q → p := λ Hp : p, λ Hq : q, Hp reveal t1 print t1 -- END
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import Lean open Lean open Lean.Elab open Lean.Elab.Term def getCtors (c : Name) : TermElabM (List Name) := do let env ← getEnv; (match env.find? c with | some (ConstantInfo.inductInfo val) => pure val.ctors | _ => pure []) def elabAnonCtor (args : Syntax) (τ : Expr) : TermElabM Expr := match τ.getAppFn with ...
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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 Mathlib.PrePort import Mathlib.Lean3Lib.init.default import Mathlib.order.category.PartialOrder import Mathlib.PostPort universes u_1 namespace Mathlib /-!...
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/- Copyright (c) 2020 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Leonardo de Moura, Sebastian Ullrich -/ prelude import Init.Core universe u v w @[reducible] def Functor.mapRev {f : Type u → Type v} [Functor f] {α β : Type u} : f α → (α → β)...
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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, Robert Y. Lewis -/ import ring_theory.witt_vector.structure_polynomial /-! # Witt vectors In this file we define the type of `p`-typical Witt vectors and ring operat...
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/- Copyright (c) 2022 Markus Himmel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Markus Himmel -/ import category_theory.subobject.well_powered import category_theory.limits.preserves.finite import category_theory.limits.shapes.finite_limits /-! # Subobjects in the...
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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, Scott Morrison, Simon Hudon Definition and basic properties of endomorphisms and automorphisms of an object in a category. -/ import Mathlib.PrePort import Mathlib.L...
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λ (m n : mynat), (mynat.rec ⟨λ (m : mynat), eq.rec (eq.refl m) (eq.rec (eq.refl (m = add zero m)) (eq.rec (eq.refl (m = add zero m)) ((mynat.rec ⟨eq.refl zero, punit.star⟩ (λ (n : mynat) (ih : pprod (add zero n = n) (mynat.rec punit (λ (n : mynat) (ih : Type), pprod (pprod (add zero n = n) ih) punit) n)), ⟨eq.rec (eq.r...
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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: Scott Morrison import category_theory.fully_faithful /- This is exactly analogous to the full_subcategory definition for a subtype, but for a sigma type instead. -/ namespace c...
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import .love01_definitions_and_statements_demo /- # LoVe Demo 3: Forward Proofs When developing a proof, often it makes sense to work __forward__: to start with what we already know and proceed step by step towards our goal. Lean's structured proofs and raw proof terms are two style that support forward reasoning. -...
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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 -/ import data.fintype.basic import data.finset.sort import group_theory.perm.basic import group_theory.order_of_element import tactic.norm_swap /-! # Sign of a permutation...
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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: Mario Carneiro -/ import data.list.nodup import data.multiset.bind import data.multiset.range /-! # The `nodup` predicate for multisets without duplicate elements. > THIS FILE...
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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.function init.data.option.basic init.util import init.category.combinators init.category.monad init.category.alternative import init.dat...
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/- INTRODUCTION and ELIMINATION RULES -/ /- For ∀ x, P x (every x has property P) - introduction rule: assume arbitrary x, then show P x - elimination rule: *apply* a proof of ∀ x, P x, as a kind of function to a specific value of x, say k, to produce a proof of P k. -/ theorem silly : ∀ (n : ℕ), true := be...
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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, Jalex Stark, Kyle Miller -/ import combinatorics.simple_graph.adj_matrix import linear_algebra.matrix.charpoly.coeff import data.int.modeq import data.zmod.basic import...
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import Test import Lean import Lean.Expr def main : IO Unit := IO.println s!"Hello, {hello}!" #eval Lean.versionString #eval main #eval IO.appPath
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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.traversable.lemmas import logic.equiv.basic /-! # Transferring `traversable` instances along isomorphisms This file allows to transfer `traversable` instan...
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def f : nat → bool | 0 := ff | _ := tt inductive tree | leaf : nat → tree | node : nat → tree → tree → tree def mk_tree : nat → nat → tree | 0 v := tree.leaf v | (n+1) v := let t := mk_tree n v in tree.node v t t def tst : tree → nat | (tree.leaf v) := v | (tree.node v l r) := match f v with | tt := tst ...
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/- Copyright (c) 2020 Yury G. Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Yury G. Kudryashov -/ import analysis.normed_space.basic /-! # Extended norm In this file we define a structure `enorm 𝕜 V` representing an extended norm (i.e., a norm that can t...
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/- Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Thomas Browning, Patrick Lutz -/ import field_theory.normal import field_theory.primitive_element import field_theory.fixed import ring_theory.power_basis import group_...
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/- Copyright (c) 2014 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov -/ import data.set.basic import logic.function.conjugate /-! # Functions over sets ## Main definitions ### Pr...
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theorem ex : let x := 1; x < 2 := by intro x decide!
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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: Scott Morrison, Reid Barton, Mario Carneiro import category_theory.limits.shape open category_theory namespace category_theory.limits universes u v w variables {C : Type u} ...
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/- LoVe Homework 6: Monads -/ import .lovelib namespace LoVe /- Question 2 **optional**: Monadic Structure on Lists -/ /- `list` can be seen as a monad, similar to `option` but with several possible outcomes. It is also similar to `set`, but the results are ordered and finite. The code below sets `list` up as a mo...
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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 Helper tactic for constructing a has_reflect instance. -/ prelude import init.meta.rec_util namespace tactic open expr environment list /- Retrieve the name...
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import standard using tactic theorem tst (a b : Prop) (H : a ↔ b) : b ↔ a := by apply iff_intro; apply (assume Hb, iff_mp_right H Hb); apply (assume Ha, iff_mp_left H Ha) check tst
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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 algebra.category.Group.Z_Module_equivalence import algebra.category.Group.limits import algebra.category.Group.colimits import algebra.category.Module.abelian im...
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import test.slim_check.testable namespace tactic.interactive open tactic slim_check meta def expect_failure (cmd : itactic) : tactic unit := λ s, match cmd s with | (interaction_monad.result.exception msg _ s') := match msg with | (some msg') := (trace (msg' ()) >> admit) s' | none := admit s' end | (inter...
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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: Scott Morrison -/ import data.buffer.parser import tactic.doc_commands open lean.parser tactic interactive parser /-- `restate_axiom` takes a structure field, and makes a new, defini...
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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 Various multiplicative and additive structures. -/ import tactic.interactive data.option section pending_1857 /- Transport multiplicative to additive...
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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 algebra.quadratic_discriminant import analysis.special_functions.pow import tactic.monotonicity /-! # Inner Product Space This file defines real inner product...
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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.tactic init.meta.rewrite_tactic init.meta.simp_tactic namespace tactic namespace interactive namespace types /- The parser treats ...
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/- Copyright (c) 2020 Anne Baanen. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Anne Baanen -/ import algebra.algebra.basic /-! # Subfields Let `K` be a field. This file defines the "bundled" subfield type `subfield K`, a type whose terms correspond to subfields o...
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open tactic example (a b c : nat) : a = c → a + b = b + c := by do n1 ← get_unused_name `a, n2 ← get_unused_name `b, n3 ← get_unused_name `c, n4 ← get_unused_name `d, intro n1, n5 ← get_unused_name `a, trace n1 >> trace n2 >> trace n3 >> trace n4 >> trace n5, get_local `a >>= subst, `[apply nat.add_c...
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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: Johan Commelin, Scott Morrison -/ import algebraic_geometry.prime_spectrum.basic import algebra.category.CommRing.colimits import algebra.category.CommRing.limits import topology.sheav...
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import tools.super open super tactic monad example (a b : ℕ → Prop) (h : ∀x, (¬a x → b x) ∧ ¬b x ∧ ¬a x) := by do c ← get_local `h >>= clause.of_classical_proof, trace c, cs ← get_clauses_classical [c], trace cs, c0 ← returnopt $ cs^.nth 0, c1 ← returnopt $ cs^.nth 1, c2 ← returnopt $ cs^.nth 2, trace c0^.type, tr...
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inductive Vec (α : Type u) : Nat → Type u | nil : Vec α 0 | cons : α → {n : Nat} → Vec α n → Vec α (n+1) def Vec.repeat (a : α) (n : Nat) : Vec α n := match n with | 0 => nil | n+1 => cons a («repeat» a n) instance [Inhabited α] : Inhabited (Vec α n) where default := Vec.repeat default n def Vec.map (...
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/- Change the below line to import week_2.kb_solutions.Part_A_groups_solutions (once the solutions are posted) if you want to get rid of the warning -/ import week_2.Part_A_groups /-! ## Subgroups We define the structure `subgroup G`, whose terms are subgroups of `G`. A subgroup of `G` is implemented as a subse...
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import Kenny.sheaf_on_opens universes v u open topological_space variables {X : Type u} [topological_space X] structure subpresheaf (F : presheaf.{u v} X) : Type (max u v) := (to_set : Π U : opens X, set (F U)) (res_mem_to_set : ∀ {U V : opens X} (HVU : V ⊆ U) {x : F U}, x ∈ to_set U → F.res U V HVU x ∈ to_set V) ...
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/- Copyright (c) 2021 Yury Kudryashov. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Yury Kudryashov -/ import topology.metric_space.metric_separated import measure_theory.constructions.borel_space import measure_theory.measure.lebesgue import analysis.special_functio...
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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.Tactic.Subst import Lean.Meta.Tactic.Clear namespace Lean namespace Meta structure CaseValueSubgoal := (mvarId : MVarId) (newH : FVarId...
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/- Copyright (c) 2017 Daniel Selsam. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Daniel Selsam Miscellaneous tactics. -/ import .util namespace tactic meta def simp_lemmas.append_simp : simp_lemmas → list name → tactic simp_lemmas | sls [] := return sls | sls...
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/- Copyright (c) 2015 Jeremy Avigad. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Robert Y. Lewis -/ import algebra.order.ring import algebra.group_power.ring /-! # Lemmas about the interaction of power operations with order Note that some lemmas are...
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import category_theory.currying import category_theory.preadditive.additive_functor import topology.category.Profinite import topology.algebra.group_completion import topology.metric_space.completion import for_mathlib.normed_group_hom_completion import locally_constant.SemiNormedGroup /- # Completions of normed gr...
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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, Jeremy Avigad, Floris van Doorn import logic.eq logic.heq data.unit structure sigma {A : Type} (B : A → Type) := dpair :: (dpr1 : A) (dpr2 : B dpr1) no...
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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, Johannes Hölzl Ordered monoids and groups. -/ import algebra.group order.bounded_lattice tactic.basic universe u variable {α : Type u} section old_structure_cmd set...
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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.Util.CollectLevelParams import Lean.Util.Recognizers import Lean.Compiler.ExternAttr import Lean.Meta.Check import Lean.Meta.Closure import Lean...
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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, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen -/ import algebra.algebra.tower import ring_theory.localization.basic /-! # Fraction ring / fraction field Frac(R)...
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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: Mario Carneiro -/ import data.list.perm import data.list.prod_monoid /-! # Multisets These are implemented as the quotient of a list by permutations. ## Notation We define the ...
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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: Scott Morrison open tactic open smt_tactic open lean.parser open interactive -- TODO raise an issue about note and pose remaining in smt_tactic meta def unfold_coe : tactic un...
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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, Yaël Dillies -/ import order.complete_lattice import order.directed /-! # Frames, completely distributive lattices and Boolean algebras In this file we define and pro...
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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, Patrick Massot -/ import group_theory.quotient_group import order.filter.pointwise import topology.algebra.monoid import topology.homeomorph import topo...
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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 Theorems that require decidability hypotheses are in the namespace "decidable". Classical versions are in the namespace "classical". Note: in the pres...
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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: Reid Barton, Scott Morrison -/ import category_theory.fin_category import category_theory.limits.cones import category_theory.adjunction.basic import category_theory.category.preorder imp...
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/- Copyright (c) 2021 Paula Neeley. All rights reserved. Author: Paula Neeley Following the textbook "Dynamic Epistemic Logic" by Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi -/ -- Def 2.1, pg 12 inductive form (agents : Type) : Type | bot : form | var (n : nat) : form | and (φ ...
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import xenalib.M1Fstuff algebra.group_power xenalib.square_root xenalib.complex local infix ` ^ ` := monoid.pow -- Need to start off with some fake reals to do Q1,2 constant fake_reals : Type @[instance] constant fake_reals_comm_ring : comm_ring fake_reals @[instance] constant fake_reals_have_lt : has_lt fake_reals...