data.nat.choose.vandermonde
⟷
Mathlib.Data.Nat.Choose.Vandermonde
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -3,7 +3,7 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
-import Data.Polynomial.Coeff
+import Algebra.Polynomial.Coeff
import Data.Nat.Choose.Basic
#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,8 +3,8 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
-import Mathbin.Data.Polynomial.Coeff
-import Mathbin.Data.Nat.Choose.Basic
+import Data.Polynomial.Coeff
+import Data.Nat.Choose.Basic
#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,15 +2,12 @@
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module data.nat.choose.vandermonde
-! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Data.Polynomial.Coeff
import Mathbin.Data.Nat.Choose.Basic
+#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"69c6a5a12d8a2b159f20933e60115a4f2de62b58"
+
/-!
# Vandermonde's identity
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -40,7 +40,6 @@ theorem Nat.add_choose_eq (m n k : ℕ) :
(m + n).choose k = ((X + 1) ^ (m + n)).coeff k := _
_ = ((X + 1) ^ m * (X + 1) ^ n).coeff k := by rw [pow_add]
_ = ∑ ij : ℕ × ℕ in antidiagonal k, m.choose ij.1 * n.choose ij.2 := _
-
· rw [coeff_X_add_one_pow, Nat.cast_id]
· rw [coeff_mul, Finset.sum_congr rfl]
simp only [coeff_X_add_one_pow, Nat.cast_id, eq_self_iff_true, imp_true_iff]
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -27,7 +27,7 @@ https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Algebraic_proof .
-/
-open BigOperators
+open scoped BigOperators
open Polynomial Finset.Nat
mathlib commit https://github.com/leanprover-community/mathlib/commit/2196ab363eb097c008d4497125e0dde23fb36db2
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
! This file was ported from Lean 3 source module data.nat.choose.vandermonde
-! leanprover-community/mathlib commit d6814c584384ddf2825ff038e868451a7c956f31
+! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.Data.Nat.Choose.Basic
# Vandermonde's identity
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
In this file we prove Vandermonde's identity (`nat.add_choose_eq`):
`(m + n).choose k = ∑ (ij : ℕ × ℕ) in antidiagonal k, m.choose ij.1 * n.choose ij.2`
mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba
@@ -28,6 +28,7 @@ open BigOperators
open Polynomial Finset.Nat
+#print Nat.add_choose_eq /-
/-- Vandermonde's identity -/
theorem Nat.add_choose_eq (m n k : ℕ) :
(m + n).choose k = ∑ ij : ℕ × ℕ in antidiagonal k, m.choose ij.1 * n.choose ij.2 :=
@@ -41,4 +42,5 @@ theorem Nat.add_choose_eq (m n k : ℕ) :
· rw [coeff_mul, Finset.sum_congr rfl]
simp only [coeff_X_add_one_pow, Nat.cast_id, eq_self_iff_true, imp_true_iff]
#align nat.add_choose_eq Nat.add_choose_eq
+-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Data
(#11751)
Polynomial
and MvPolynomial
are algebraic objects, hence should be under Algebra
(or at least not under Data
)
@@ -3,7 +3,7 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
-import Mathlib.Data.Polynomial.Coeff
+import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Data.Nat.Choose.Basic
#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
Finset.Nat.antidiagonal
(#7486)
We define a type class Finset.HasAntidiagonal A
which contains a function
antidiagonal : A → Finset (A × A)
such that antidiagonal n
is the Finset of all pairs adding to n
, as witnessed by mem_antidiagonal
.
When A
is a canonically ordered add monoid with locally finite order
this typeclass can be instantiated with Finset.antidiagonalOfLocallyFinite
.
This applies in particular when A
is ℕ
, more generally or σ →₀ ℕ
,
or even ι →₀ A
under the additional assumption OrderedSub A
that make it a canonically ordered add monoid.
(In fact, we would just need an AddMonoid
with a compatible order,
finite Iic
, such that if a + b = n
, then a, b ≤ n
,
and any finiteness condition would be OK.)
For computational reasons it is better to manually provide instances for ℕ
and σ →₀ ℕ
, to avoid quadratic runtime performance.
These instances are provided as Finset.Nat.instHasAntidiagonal
and Finsupp.instHasAntidiagonal
.
This is why Finset.antidiagonalOfLocallyFinite
is an abbrev
and not an instance
.
This definition does not exactly match with that of Multiset.antidiagonal
defined in Mathlib.Data.Multiset.Antidiagonal
, because of the multiplicities.
Indeed, by counting multiplicities, Multiset α
is equivalent to α →₀ ℕ
,
but Finset.antidiagonal
and Multiset.antidiagonal
will return different objects.
For example, for s : Multiset ℕ := {0,0,0}
, Multiset.antidiagonal s
has 8 elements
but Finset.antidiagonal s
has only 4.
def s : Multiset ℕ := {0, 0, 0}
#eval (Finset.antidiagonal s).card -- 4
#eval Multiset.card (Multiset.antidiagonal s) -- 8
HasMulAntidiagonal
(for monoids).
For PNat
, we will recover the set of divisors of a strictly positive integer.This closes #7917
Co-authored by: María Inés de Frutos-Fernández <mariaines.dff@gmail.com> and Eric Wieser <efw27@cam.ac.uk>
Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@@ -23,7 +23,7 @@ https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Algebraic_proof .
open BigOperators
-open Polynomial Finset.Nat
+open Polynomial Finset Finset.Nat
/-- Vandermonde's identity -/
theorem Nat.add_choose_eq (m n k : ℕ) :
@@ -2,15 +2,12 @@
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module data.nat.choose.vandermonde
-! leanprover-community/mathlib commit d6814c584384ddf2825ff038e868451a7c956f31
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Data.Polynomial.Coeff
import Mathlib.Data.Nat.Choose.Basic
+#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
+
/-!
# Vandermonde's identity
fix-comments.py
on all files.@@ -15,7 +15,7 @@ import Mathlib.Data.Nat.Choose.Basic
# Vandermonde's identity
-In this file we prove Vandermonde's identity (`nat.add_choose_eq`):
+In this file we prove Vandermonde's identity (`Nat.add_choose_eq`):
`(m + n).choose k = ∑ (ij : ℕ × ℕ) in antidiagonal k, m.choose ij.1 * n.choose ij.2`
We follow the algebraic proof from
The unported dependencies are