5b1b78a8 - mathematics_in

Authored by: Patrick Massot
Authored at: 2024-11-11 10:26:08 +0100
Committed by: Patrick Massot
Committed at: 2024-11-11 10:26:08 +0100
Object name: 5b1b78a8c12b187e81c33c483e316c243d17d237
Parent: 45b50b91

Update lun. 11 nov. 2024 10:26:08 CET

Changes

16 changed files (+102/-89)

MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean
- @@ -102,25 +102,25 @@ example : 0 ≤ a ^ 2 := by
  example (h : a ≤ b) : c - exp b ≤ c - exp a := by sorry - example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + example : 2*a*b ≤ a^2 + b^2 := by + have h : 0 ≤ a^2 - 2*a*b + b^2 calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + a^2 - 2*a*b + b^2 = (a - b)^2 := by ring _ ≥ 0 := by apply pow_two_nonneg calc - 2 * a * b = 2 * a * b + 0 := by ring - _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := add_le_add (le_refl _) h - _ = a ^ 2 + b ^ 2 := by ring + 2*a*b = 2*a*b + 0 := by ring + _ ≤ 2*a*b + (a^2 - 2*a*b + b^2) := add_le_add (le_refl _) h + _ = a^2 + b^2 := by ring - example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + example : 2*a*b ≤ a^2 + b^2 := by + have h : 0 ≤ a^2 - 2*a*b + b^2 calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + a^2 - 2*a*b + b^2 = (a - b)^2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith - example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by + example : |a*b| ≤ (a^2 + b^2)/2 := by sorry #check abs_le'.mpr
MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean
- @@ -35,21 +35,21 @@ example (h : a ≤ b) : c - exp b ≤ c - exp a := by
 example (h : a ≤ b) : c - exp b ≤ c - exp a := by linarith [exp_le_exp.mpr h] - theorem fact1 : a * b * 2 ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + theorem fact1 : a*b*2 ≤ a^2 + b^2 := by + have h : 0 ≤ a^2 - 2*a*b + b^2 calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + a^2 - 2*a*b + b^2 = (a - b)^2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith - theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 + theorem fact2 : -(a*b)*2 ≤ a^2 + b^2 := by + have h : 0 ≤ a^2 + 2*a*b + b^2 calc - a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring + a^2 + 2*a*b + b^2 = (a + b)^2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith - example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by + example : |a*b| ≤ (a^2 + b^2)/2 := by have h : (0 : ℝ) < 2 := by norm_num apply abs_le'.mpr constructor
MIL/C08_Groups_and_Rings/S02_Rings.lean
- @@ -1,4 +1,4 @@
  - import Mathlib.RingTheory.Ideal.QuotientOperations + import Mathlib.RingTheory.Ideal.Quotient.Operations import Mathlib.RingTheory.Localization.Basic import Mathlib.RingTheory.DedekindDomain.Ideal import Mathlib.Analysis.Complex.Polynomial.Basic
- @@ -211,9 +211,9 @@ example : aroots (X ^ 2 + 1 : ℝ[X]) ℂ = {Complex.I, -I} := by
  -- Mathlib knows about D'Alembert-Gauss theorem: ``ℂ`` is algebraically closed. example : IsAlgClosed ℂ := inferInstance - #check (Complex.ofReal : ℝ →+* ℂ) + #check (Complex.ofRealHom : ℝ →+* ℂ) - example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp + example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofRealHom Complex.I = 0 := by simp open MvPolynomial
MIL/C08_Groups_and_Rings/solutions/Solutions_S02_Rings.lean
- @@ -1,4 +1,4 @@
  - import Mathlib.RingTheory.Ideal.QuotientOperations + import Mathlib.RingTheory.Ideal.Quotient.Operations import Mathlib.RingTheory.Localization.Basic import Mathlib.RingTheory.DedekindDomain.Ideal import Mathlib.Analysis.Complex.Polynomial.Basic
MIL/C09_Linear_Algebra/S02_Subspaces.lean
- @@ -100,11 +100,15 @@ example : GaloisInsertion (Submodule.span K) ((↑) : Submodule K V → Set V) :
  example {S T : Submodule K V} {x : V} (h : x ∈ S ⊔ T) : ∃ s ∈ S, ∃ t ∈ T, x = s + t := by rw [← S.span_eq, ← T.span_eq, ← Submodule.span_union] at h - apply Submodule.span_induction h (p := fun x ↦ ∃ s ∈ S, ∃ t ∈ T, x = s + t) - · sorry - · sorry - · sorry - · sorry + induction h using Submodule.span_induction with + | mem y h => + sorry + | zero => + sorry + | add x y hx hy hx' hy' => + sorry + | smul a x hx hx' => + sorry section
MIL/C09_Linear_Algebra/S03_Endomorphisms.lean
- @@ -42,11 +42,8 @@ example (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊓ k
  example (P Q : K[X]) (h : IsCoprime P Q) (φ : End K V) : ker (aeval φ P) ⊔ ker (aeval φ Q) = ker (aeval φ (P*Q)) := by sorry - example (a : K) : algebraMap K (End K V) a = a • LinearMap.id := rfl - - example (φ : End K V) (a : K) : - φ.eigenspace a = LinearMap.ker (φ - algebraMap K (End K V) a) := - rfl + example (φ : End K V) (a : K) : φ.eigenspace a = LinearMap.ker (φ - a • 1) := + End.eigenspace_def
MIL/C09_Linear_Algebra/solutions/Solutions_S02_Subspaces.lean
- @@ -21,19 +21,24 @@ def preimage {W : Type*} [AddCommGroup W] [Module K W] (φ : V →ₗ[K] W) (H :
  rw [Set.mem_preimage, map_smul] exact H.smul_mem a hv - · rintro x (hx|hx) - · use x, hx, 0, T.zero_mem + | mem x h => + rcases h with (hx|hx) + · use x, hx, 0, T.zero_mem + module + · use 0, S.zero_mem, x, hx + module + | zero => + use 0, S.zero_mem, 0, T.zero_mem module - · use 0, S.zero_mem, x, hx + | add x y hx hy hx' hy' => + rcases hx' with ⟨s, hs, t, ht, rfl⟩ + rcases hy' with ⟨s', hs', t', ht', rfl⟩ + use s + s', S.add_mem hs hs', t + t', T.add_mem ht ht' + module + | smul a x hx hx' => + rcases hx' with ⟨s, hs, t, ht, rfl⟩ + use a • s, S.smul_mem a hs, a • t, T.smul_mem a ht module - · use 0, S.zero_mem, 0, T.zero_mem - module - · rintro - - ⟨s, hs, t, ht, rfl⟩ ⟨s', hs', t', ht', rfl⟩ - use s + s', S.add_mem hs hs', t + t', T.add_mem ht ht' - module - · rintro a - ⟨s, hs, t, ht, rfl⟩ - use a • s, S.smul_mem a hs, a • t, T.smul_mem a ht - module constructor · intro h x hx
MIL/C10_Topology/solutions/Solutions_S01_Filters.lean
- @@ -65,7 +65,7 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :
  Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by rw [nhds_prod_eq] unfold Tendsto SProd.sprod Filter.instSProd Filter.prod - erw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M :=
html/C02_Basics.html
- @@ -930,16 +930,16 @@ next example:
 Using the same tricks, confirm that <code class="docutils literal notranslate">linarith</code> instead of <code class="docutils literal notranslate">apply?</code> can also finish the job. Here is another example of an inequality: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 2*a*b ≤ a^2 + b^2 := by + have h : 0 ≤ a^2 - 2*a*b + b^2 calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + a^2 - 2*a*b + b^2 = (a - b)^2 := by ring _ ≥ 0 := by apply pow_two_nonneg calc - 2 * a * b = 2 * a * b + 0 := by ring - _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := add_le_add (le_refl _) h - _ = a ^ 2 + b ^ 2 := by ring + 2*a*b = 2*a*b + 0 := by ring + _ ≤ 2*a*b + (a^2 - 2*a*b + b^2) := add_le_add (le_refl _) h + _ = a^2 + b^2 := by ring </pre></div> </div> Mathlib tends to put spaces around binary operations like <code class="docutils literal notranslate">*</code> and <code class="docutils literal notranslate">^</code>,
- @@ -959,10 +959,10 @@ we can simply write the proof term <code class="docutils literal notranslate"><s
 out the hypothesis <code class="docutils literal notranslate">h</code>. Once we have it, the second calculation involves only linear arithmetic, and <code class="docutils literal notranslate">linarith</code> can handle it: - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by - have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : 2*a*b ≤ a^2 + b^2 := by + have h : 0 ≤ a^2 - 2*a*b + b^2 calc - a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring + a^2 - 2*a*b + b^2 = (a - b)^2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith </pre></div>
- @@ -971,7 +971,7 @@ linear arithmetic, and <code class="docutils literal notranslate">abs_le'.mpr</code>. You will also need the <code class="docutils literal notranslate">constructor</code> tactic to split a conjunction to two goals; see <a class="reference internal" href="C03_Logic.html#conjunction-and-biimplication">Section 3.4</a>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by + <div class="highlight-lean notranslate"><div class="highlight"><pre>example : |a*b| ≤ (a^2 + b^2)/2 := by sorry #check abs_le'.mpr
- @@ -1359,7 +1359,7 @@ either distributivity law implies the other:
 </pre></div> </div> It is possible to combine axiomatic structures into larger ones. - For example, a strict ordered ring consists of a commutative ring together + For example, a strict ordered ring consists of a ring together with a partial order on the carrier satisfying additional axioms that say that the ring operations are compatible with the order:
- @@ -1380,7 +1380,8 @@ used to reason about arithmetic and the ordering on the real
 numbers hold generically for any ordered ring. Here are a couple of examples you can try, using only properties of rings, partial orders, and the facts - enumerated in the last two examples: + enumerated in the last two examples (beware that those rings are + not assumed to be commutative, so the <cite>ring</cite> tactic is not available): <div class="highlight-lean notranslate"><div class="highlight"><pre>example (h : a ≤ b) : 0 ≤ b - a := by sorry
html/C08_Groups_and_Rings.html
- @@ -1087,9 +1087,9 @@ for <code class="docutils literal notranslate">roots</c
 in <code class="docutils literal notranslate">S</code> using <code class="docutils literal notranslate">Polynomial.eval₂</code>. This one produces an actual function from <code class="docutils literal notranslate">R[X]</code> to <code class="docutils literal notranslate">S</code> since it does not assume the existence of a <code class="docutils literal notranslate">Algebra R S</code> instance, so dot notation works as you would expect. - <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (Complex.ofReal : ℝ →+* ℂ) + <div class="highlight-lean notranslate"><div class="highlight"><pre>#check (Complex.ofRealHom : ℝ →+* ℂ) - example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofReal Complex.I = 0 := by simp + example : (X ^ 2 + 1 : ℝ[X]).eval₂ Complex.ofRealHom Complex.I = 0 := by simp </pre></div> </div> Let us end by mentioning multivariate polynomials briefly. Given a commutative semiring <code class="docutils literal notranslate">R</code>,
html/C09_Linear_Algebra.html
- @@ -540,19 +540,22 @@ and the whole theory of Galois connections.
 When those are not enough, one can use the relevant induction principle <code class="docutils literal notranslate">Submodule.span_induction</code> which ensures a property holds for every element of the span of <code class="docutils literal notranslate">s</code> as long as it holds on <code class="docutils literal notranslate">zero</code> and elements of <code class="docutils literal notranslate">s</code> and is stable under - sum and scalar multiplication. As usual with such lemmas, Lean sometimes needs help - to figure out the predicate we are interested in. + sum and scalar multiplication. As an exercise, let us reprove one implication of <code class="docutils literal notranslate">Submodule.mem_sup</code>. Remember that you can use the <cite>module</cite> tactic to close goals that follow from the axioms relating the various algebraic operations on <code class="docutils literal notranslate">V</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>example {S T : Submodule K V} {x : V} (h : x ∈ S ⊔ T) : ∃ s ∈ S, ∃ t ∈ T, x = s + t := by rw [← S.span_eq, ← T.span_eq, ← Submodule.span_union] at h - apply Submodule.span_induction h (p := fun x ↦ ∃ s ∈ S, ∃ t ∈ T, x = s + t) - · sorry - · sorry - · sorry - · sorry + induction h using Submodule.span_induction with + | mem y h => + sorry + | zero => + sorry + | add x y hx hy hx' hy' => + sorry + | smul a x hx hx' => + sorry </pre></div> </div> </section>
- @@ -694,21 +697,14 @@ prime polynomials \(P\) and <s
 sorry </pre></div> </div> - We now move to the discussions of eigenspaces and eigenvalues. By the definition, the eigenspace + We now move to the discussions of eigenspaces and eigenvalues. The eigenspace associated to an endomorphism \(φ\) and a scalar \(a\) is the kernel of \(φ - aId\). - The first thing to understand is how to spell \(aId\). We could use - <code class="docutils literal notranslate">a • LinearMap.id</code>, but Mathlib uses <cite>algebraMap K (End K V) a</cite> which directly plays nicely - with the <code class="docutils literal notranslate">K</code>-algebra structure. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (a : K) : algebraMap K (End K V) a = a • LinearMap.id := rfl - </pre></div> - </div> - Then the next thing to note is that eigenspaces are defined for all values of <code class="docutils literal notranslate">a</code>, although + Eigenspaces are defined for all values of <code class="docutils literal notranslate">a</code>, although they are interesting only when they are non-zero. However an eigenvector is, by definition, a non-zero element of an eigenspace. The corresponding predicate is <code class="docutils literal notranslate">End.HasEigenvector</code>. - <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ : End K V) (a : K) : - φ.eigenspace a = LinearMap.ker (φ - algebraMap K (End K V) a) := - rfl + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (φ : End K V) (a : K) : φ.eigenspace a = LinearMap.ker (φ - a • 1) := + End.eigenspace_def </pre></div> </div> Then there is a predicate <code class="docutils literal notranslate">End.HasEigenvalue</code> and the corresponding subtype <code class="docutils literal notranslate">End.Eigenvalues</code>.
html/searchindex.js
- @@ -1,1 +1,1 @@
 - Search.setIndex({"alltitles": {"Algebraic Structures": [[5, "algebraic-structures"]], "Algebras and polynomials": [[7, "algebras-and-polynomials"]], "Asymptotic comparisons": [[10, "asymptotic-comparisons"]], "Balls, open sets and closed sets": [[9, "balls-open-sets-and-closed-sets"]], "Bases": [[8, "bases"]], "Basics": [[1, null], [6, "basics"]], "Building the Gaussian Integers": [[5, "building-the-gaussian-integers"]], "Calculating": [[1, "calculating"]], "Compactness": [[9, "compactness"], [9, "id5"]], "Complete lattice structure and internal direct sums": [[8, "complete-lattice-structure-and-internal-direct-sums"]], "Completeness": [[9, "completeness"]], "Concrete groups": [[7, "concrete-groups"]], "Conjunction and Iff": [[2, "conjunction-and-iff"]], "Continuous linear maps": [[10, "continuous-linear-maps"]], "Convergence and continuity": [[9, "convergence-and-continuity"]], "Defining structures": [[5, "defining-structures"]], "Differentiability": [[10, "differentiability"]], 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"titles": ["1. Introduction", "2. Basics", "3. Logic", "4. Sets and Functions", "5. Elementary Number Theory", "6. Structures", "7. Hierarchies", "8. Groups and Rings", "9. Linear algebra", "10. Topology", "11. Differential Calculus", "12. Integration and Measure Theory", "Index", "Mathematics in Lean"], "titleterms": {"The": [2, 3], "about": 1, "action": 7, "algebra": [1, 5, 7, 8], "appli": 1, "asymptot": 10, "ball": 9, "base": 8, "basic": [1, 6], "bernstein": 3, "build": 5, "calcul": 1, "calculu": 10, "close": 9, "compact": 9, "comparison": 10, "complet": [8, 9], "concret": 7, "conjunct": 2, "continu": [9, 10], "converg": [2, 9], "countabl": 9, "defin": 5, "differenti": 10, "dimens": 8, "direct": 8, "disjunct": 2, "elementari": [4, 10, 11], "endomorph": 8, "exampl": 1, "existenti": 2, "fact": 1, "filter": 9, "function": [3, 9], "fundament": 9, "gaussian": 5, "get": 0, "group": 7, "hierarchi": 6, "ideal": 7, "ident": 1, "iff": 2, "implic": 2, "index": 12, "induct": 4, "infinit": 4, "integ": 5, "integr": 11, "intern": 8, "introduct": 0, "irrat": 4, "lattic": 8, "lean": 13, "lemma": 1, "linear": [8, 10], "logic": 2, "mani": 4, "map": [8, 10], "mathemat": 13, "matric": 8, "measur": 11, "metric": 9, "monoid": 7, "more": 1, "morphism": [6, 7], "negat": 2, "norm": 10, "number": 4, "object": 6, "open": 9, "overview": 0, "polynomi": 7, "prime": 4, "product": 8, "prove": 1, "pull": 8, "push": 8, "quantifi": 2, "quotient": [7, 8], "recurs": 4, "ring": 7, "root": 4, "rw": 1, "schr\u00f6der": 3, "separ": 9, "sequenc": 2, "set": [3, 8, 9], "space": [8, 9, 10], "span": 8, "start": 0, "structur": [1, 5, 8], "sub": 6, "subgroup": 7, "subr": 7, "subspac": 8, "sum": 8, "theorem": [1, 3], "theori": [4, 11], "topolog": 9, "topologi": 9, "uniformli": 9, "unit": 7, "univers": 2, "us": 1, "vector": 8}})
lake-manifest.json
- @@ -5,17 +5,17 @@
  "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "13f9b00769bdac2c0041406a6c2524a361e8d660", + "rev": "31a10a332858d6981dbcf55d54ee51680dd75f18", "name": "batteries", "manifestFile": "lake-manifest.json", "inputRev": "main", "inherited": true, - "configFile": "lakefile.lean"}, + "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/quote4", "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "2c8ae451ce9ffc83554322b14437159c1a9703f9", + "rev": "1357f4f49450abb9dfd4783e38219f4ce84f9785", "name": "Qq", "manifestFile": "lake-manifest.json", "inputRev": "master",
- @@ -25,7 +25,7 @@
  "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "b20a88676fd00affb90cbc9f1ff004ae588103b3", + "rev": "5f934891e11d70a1b86e302fdf9cecfc21e8de46", "name": "aesop", "manifestFile": "lake-manifest.json", "inputRev": "master",
- @@ -35,16 +35,16 @@
  "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "eb08eee94098fe530ccd6d8751a86fe405473d4c", + "rev": "23268f52d3505955de3c26a42032702c25cfcbf8", "name": "proofwidgets", "manifestFile": "lake-manifest.json", - "inputRev": "v0.0.42", + "inputRev": "v0.0.44", "inherited": true, "configFile": "lakefile.lean"}, {"url": "https://github.com/leanprover/lean4-cli", "type": "git", "subDir": null, - "scope": "", + "scope": "leanprover", "rev": "2cf1030dc2ae6b3632c84a09350b675ef3e347d0", "name": "Cli", "manifestFile": "lake-manifest.json",
- @@ -55,7 +55,7 @@
  "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "63a7d4a353f48f6c5f1bc19d0f018b0513cb370a", + "rev": "984d7ee170b75d6b03c0903e0b750ee2c6d1e3fb", "name": "importGraph", "manifestFile": "lake-manifest.json", "inputRev": "main",
- @@ -65,20 +65,30 @@
  "type": "git", "subDir": null, "scope": "leanprover-community", - "rev": "781beceb959c68b36d3d96205b3531e341879d2c", + "rev": "7bedaed1ef024add1e171cc17706b012a9a37802", "name": "LeanSearchClient", "manifestFile": "lake-manifest.json", "inputRev": "main", "inherited": true, "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover-community/plausible", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "d212dd74414e997653cd3484921f4159c955ccca", + "name": "plausible", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml"}, {"url": "https://github.com/leanprover-community/mathlib4", "type": "git", "subDir": null, "scope": "", - "rev": "5e050d47562fa4938a5f9afbc006c7f02f4544aa", + "rev": "d7317655e2826dc1f1de9a0c138db2775c4bb841", "name": "mathlib", "manifestFile": "lake-manifest.json", - "inputRev": "master", + "inputRev": "v4.13.0", "inherited": false, "configFile": "lakefile.lean"}], "name": "mil",
lakefile.lean
- @@ -10,5 +10,5 @@ package mil where
  @[default_target] lean_lib MIL where - require mathlib from git "https://github.com/leanprover-community/mathlib4"@"master" + require mathlib from git "https://github.com/leanprover-community/mathlib4"@"v4.13.0"
lean-toolchain
- @@ -1,1 +1,1 @@
  - leanprover/lean4:v4.13.0-rc3 + leanprover/lean4:v4.13.0
mathematics_in_lean.pdf