Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
variable {α β G M : Type*}
@[to_additive]
instance CommMagma.to_isCommutative [CommMagma G] : Std.Commutative (α := G) (· * ·) := ⟨mul_comm⟩
#align comm_semigroup.to_is_commutative CommMagma.to_isCommutative
#align add_comm_semigroup.to_is_commutative AddCommMagma.to_isCommutative
attribute [local simp] mul_assoc sub_eq_add_neg
section DivInvMonoid
variable [DivInvMonoid G] {a b c : G}
@[to_additive, field_simps] -- The attributes are out of order on purpose
theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by rw [div_eq_mul_inv, one_mul]
#align inv_eq_one_div inv_eq_one_div
#align neg_eq_zero_sub neg_eq_zero_sub
@[to_additive]
| Mathlib/Algebra/Group/Basic.lean | 450 | 451 | theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by |
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
| 147 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
variable {α β G M : Type*}
@[to_additive]
instance CommMagma.to_isCommutative [CommMagma G] : Std.Commutative (α := G) (· * ·) := ⟨mul_comm⟩
#align comm_semigroup.to_is_commutative CommMagma.to_isCommutative
#align add_comm_semigroup.to_is_commutative AddCommMagma.to_isCommutative
attribute [local simp] mul_assoc sub_eq_add_neg
section DivInvMonoid
variable [DivInvMonoid G] {a b c : G}
@[to_additive, field_simps] -- The attributes are out of order on purpose
theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by rw [div_eq_mul_inv, one_mul]
#align inv_eq_one_div inv_eq_one_div
#align neg_eq_zero_sub neg_eq_zero_sub
@[to_additive]
theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
#align mul_one_div mul_one_div
#align add_zero_sub add_zero_sub
@[to_additive]
| Mathlib/Algebra/Group/Basic.lean | 456 | 457 | theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) := by |
rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]
| 147 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
variable {α β G M : Type*}
@[to_additive]
instance CommMagma.to_isCommutative [CommMagma G] : Std.Commutative (α := G) (· * ·) := ⟨mul_comm⟩
#align comm_semigroup.to_is_commutative CommMagma.to_isCommutative
#align add_comm_semigroup.to_is_commutative AddCommMagma.to_isCommutative
attribute [local simp] mul_assoc sub_eq_add_neg
section DivInvMonoid
variable [DivInvMonoid G] {a b c : G}
@[to_additive, field_simps] -- The attributes are out of order on purpose
theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by rw [div_eq_mul_inv, one_mul]
#align inv_eq_one_div inv_eq_one_div
#align neg_eq_zero_sub neg_eq_zero_sub
@[to_additive]
theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
#align mul_one_div mul_one_div
#align add_zero_sub add_zero_sub
@[to_additive]
theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) := by
rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]
#align mul_div_assoc mul_div_assoc
#align add_sub_assoc add_sub_assoc
@[to_additive, field_simps] -- The attributes are out of order on purpose
theorem mul_div_assoc' (a b c : G) : a * (b / c) = a * b / c :=
(mul_div_assoc _ _ _).symm
#align mul_div_assoc' mul_div_assoc'
#align add_sub_assoc' add_sub_assoc'
@[to_additive (attr := simp)]
theorem one_div (a : G) : 1 / a = a⁻¹ :=
(inv_eq_one_div a).symm
#align one_div one_div
#align zero_sub zero_sub
@[to_additive]
| Mathlib/Algebra/Group/Basic.lean | 474 | 474 | theorem mul_div (a b c : G) : a * (b / c) = a * b / c := by | simp only [mul_assoc, div_eq_mul_inv]
| 147 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
variable {α β G M : Type*}
@[to_additive]
instance CommMagma.to_isCommutative [CommMagma G] : Std.Commutative (α := G) (· * ·) := ⟨mul_comm⟩
#align comm_semigroup.to_is_commutative CommMagma.to_isCommutative
#align add_comm_semigroup.to_is_commutative AddCommMagma.to_isCommutative
attribute [local simp] mul_assoc sub_eq_add_neg
section DivInvMonoid
variable [DivInvMonoid G] {a b c : G}
@[to_additive, field_simps] -- The attributes are out of order on purpose
theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by rw [div_eq_mul_inv, one_mul]
#align inv_eq_one_div inv_eq_one_div
#align neg_eq_zero_sub neg_eq_zero_sub
@[to_additive]
theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
#align mul_one_div mul_one_div
#align add_zero_sub add_zero_sub
@[to_additive]
theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) := by
rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]
#align mul_div_assoc mul_div_assoc
#align add_sub_assoc add_sub_assoc
@[to_additive, field_simps] -- The attributes are out of order on purpose
theorem mul_div_assoc' (a b c : G) : a * (b / c) = a * b / c :=
(mul_div_assoc _ _ _).symm
#align mul_div_assoc' mul_div_assoc'
#align add_sub_assoc' add_sub_assoc'
@[to_additive (attr := simp)]
theorem one_div (a : G) : 1 / a = a⁻¹ :=
(inv_eq_one_div a).symm
#align one_div one_div
#align zero_sub zero_sub
@[to_additive]
theorem mul_div (a b c : G) : a * (b / c) = a * b / c := by simp only [mul_assoc, div_eq_mul_inv]
#align mul_div mul_div
#align add_sub add_sub
@[to_additive]
| Mathlib/Algebra/Group/Basic.lean | 479 | 479 | theorem div_eq_mul_one_div (a b : G) : a / b = a * (1 / b) := by | rw [div_eq_mul_inv, one_div]
| 147 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
variable {α β G M : Type*}
@[to_additive]
instance CommMagma.to_isCommutative [CommMagma G] : Std.Commutative (α := G) (· * ·) := ⟨mul_comm⟩
#align comm_semigroup.to_is_commutative CommMagma.to_isCommutative
#align add_comm_semigroup.to_is_commutative AddCommMagma.to_isCommutative
attribute [local simp] mul_assoc sub_eq_add_neg
section DivInvOneMonoid
variable [DivInvOneMonoid G]
@[to_additive (attr := simp)]
| Mathlib/Algebra/Group/Basic.lean | 490 | 490 | theorem div_one (a : G) : a / 1 = a := by | simp [div_eq_mul_inv]
| 147 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
variable {α β G M : Type*}
@[to_additive]
instance CommMagma.to_isCommutative [CommMagma G] : Std.Commutative (α := G) (· * ·) := ⟨mul_comm⟩
#align comm_semigroup.to_is_commutative CommMagma.to_isCommutative
#align add_comm_semigroup.to_is_commutative AddCommMagma.to_isCommutative
attribute [local simp] mul_assoc sub_eq_add_neg
section multiplicative
variable [Monoid β] (p r : α → α → Prop) [IsTotal α r] (f : α → α → β)
@[to_additive additive_of_symmetric_of_isTotal]
lemma multiplicative_of_symmetric_of_isTotal
(hsymm : Symmetric p) (hf_swap : ∀ {a b}, p a b → f a b * f b a = 1)
(hmul : ∀ {a b c}, r a b → r b c → p a b → p b c → p a c → f a c = f a b * f b c)
{a b c : α} (pab : p a b) (pbc : p b c) (pac : p a c) : f a c = f a b * f b c := by
have hmul' : ∀ {b c}, r b c → p a b → p b c → p a c → f a c = f a b * f b c := by
intros b c rbc pab pbc pac
obtain rab | rba := total_of r a b
· exact hmul rab rbc pab pbc pac
rw [← one_mul (f a c), ← hf_swap pab, mul_assoc]
obtain rac | rca := total_of r a c
· rw [hmul rba rac (hsymm pab) pac pbc]
· rw [hmul rbc rca pbc (hsymm pac) (hsymm pab), mul_assoc, hf_swap (hsymm pac), mul_one]
obtain rbc | rcb := total_of r b c
· exact hmul' rbc pab pbc pac
· rw [hmul' rcb pac (hsymm pbc) pab, mul_assoc, hf_swap (hsymm pbc), mul_one]
#align multiplicative_of_symmetric_of_is_total multiplicative_of_symmetric_of_isTotal
#align additive_of_symmetric_of_is_total additive_of_symmetric_of_isTotal
@[to_additive additive_of_isTotal "If a binary function from a type equipped with a total relation
`r` to an additive monoid is anti-symmetric (i.e. satisfies `f a b + f b a = 0`), in order to show
it is additive (i.e. satisfies `f a c = f a b + f b c`), we may assume `r a b` and `r b c` are
satisfied. We allow restricting to a subset specified by a predicate `p`."]
| Mathlib/Algebra/Group/Basic.lean | 1,426 | 1,432 | theorem multiplicative_of_isTotal (p : α → Prop) (hswap : ∀ {a b}, p a → p b → f a b * f b a = 1)
(hmul : ∀ {a b c}, r a b → r b c → p a → p b → p c → f a c = f a b * f b c) {a b c : α}
(pa : p a) (pb : p b) (pc : p c) : f a c = f a b * f b c := by |
apply multiplicative_of_symmetric_of_isTotal (fun a b => p a ∧ p b) r f fun _ _ => And.symm
· simp_rw [and_imp]; exact @hswap
· exact fun rab rbc pab _pbc pac => hmul rab rbc pab.1 pab.2 pac.2
exacts [⟨pa, pb⟩, ⟨pb, pc⟩, ⟨pa, pc⟩]
| 147 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Logic.Embedding.Basic
#align_import algebra.hom.embedding from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
variable {G : Type*}
section LeftOrRightCancelSemigroup
@[to_additive (attr := simps)
"If left-addition by any element is cancellative, left-addition by `g` is an
embedding."]
def mulLeftEmbedding [Mul G] [IsLeftCancelMul G] (g : G) : G ↪ G where
toFun h := g * h
inj' := mul_right_injective g
#align mul_left_embedding mulLeftEmbedding
#align add_left_embedding addLeftEmbedding
#align add_left_embedding_apply addLeftEmbedding_apply
#align mul_left_embedding_apply mulLeftEmbedding_apply
@[to_additive (attr := simps)
"If right-addition by any element is cancellative, right-addition by `g` is an
embedding."]
def mulRightEmbedding [Mul G] [IsRightCancelMul G] (g : G) : G ↪ G where
toFun h := h * g
inj' := mul_left_injective g
#align mul_right_embedding mulRightEmbedding
#align add_right_embedding addRightEmbedding
#align mul_right_embedding_apply mulRightEmbedding_apply
#align add_right_embedding_apply addRightEmbedding_apply
@[to_additive]
| Mathlib/Algebra/Group/Embedding.lean | 49 | 52 | theorem mulLeftEmbedding_eq_mulRightEmbedding [CommSemigroup G] [IsCancelMul G] (g : G) :
mulLeftEmbedding g = mulRightEmbedding g := by |
ext
exact mul_comm _ _
| 148 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : ℕ → Option ℕ
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@[simp]
theorem ppred_zero : ppred 0 = none := rfl
@[simp]
theorem ppred_succ {n : ℕ} : ppred (succ n) = some n := rfl
def psub (m : ℕ) : ℕ → Option ℕ
| 0 => some m
| n + 1 => psub m n >>= ppred
#align nat.psub Nat.psub
@[simp]
theorem psub_zero {m : ℕ} : psub m 0 = some m := rfl
@[simp]
theorem psub_succ {m n : ℕ} : psub m (succ n) = psub m n >>= ppred := rfl
| Mathlib/Data/Nat/PSub.lean | 54 | 54 | theorem pred_eq_ppred (n : ℕ) : pred n = (ppred n).getD 0 := by | cases n <;> rfl
| 149 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : ℕ → Option ℕ
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@[simp]
theorem ppred_zero : ppred 0 = none := rfl
@[simp]
theorem ppred_succ {n : ℕ} : ppred (succ n) = some n := rfl
def psub (m : ℕ) : ℕ → Option ℕ
| 0 => some m
| n + 1 => psub m n >>= ppred
#align nat.psub Nat.psub
@[simp]
theorem psub_zero {m : ℕ} : psub m 0 = some m := rfl
@[simp]
theorem psub_succ {m n : ℕ} : psub m (succ n) = psub m n >>= ppred := rfl
theorem pred_eq_ppred (n : ℕ) : pred n = (ppred n).getD 0 := by cases n <;> rfl
#align nat.pred_eq_ppred Nat.pred_eq_ppred
theorem sub_eq_psub (m : ℕ) : ∀ n, m - n = (psub m n).getD 0
| 0 => rfl
| n + 1 => (pred_eq_ppred (m - n)).trans <| by rw [sub_eq_psub m n, psub]; cases psub m n <;> rfl
#align nat.sub_eq_psub Nat.sub_eq_psub
@[simp]
theorem ppred_eq_some {m : ℕ} : ∀ {n}, ppred n = some m ↔ succ m = n
| 0 => by constructor <;> intro h <;> contradiction
| n + 1 => by constructor <;> intro h <;> injection h <;> subst m <;> rfl
#align nat.ppred_eq_some Nat.ppred_eq_some
-- Porting note: `contradiction` required an `intro` for the goals
-- `ppred (n + 1) = none → n + 1 = 0` and `n + 1 = 0 → ppred (n + 1) = none`
@[simp]
theorem ppred_eq_none : ∀ {n : ℕ}, ppred n = none ↔ n = 0
| 0 => by simp
| n + 1 => by constructor <;> intro <;> contradiction
#align nat.ppred_eq_none Nat.ppred_eq_none
theorem psub_eq_some {m : ℕ} : ∀ {n k}, psub m n = some k ↔ k + n = m
| 0, k => by simp [eq_comm]
| n + 1, k => by
apply Option.bind_eq_some.trans
simp only [psub_eq_some, ppred_eq_some]
simp [add_comm, add_left_comm, Nat.succ_eq_add_one]
#align nat.psub_eq_some Nat.psub_eq_some
| Mathlib/Data/Nat/PSub.lean | 85 | 93 | theorem psub_eq_none {m n : ℕ} : psub m n = none ↔ m < n := by |
cases s : psub m n <;> simp [eq_comm]
· show m < n
refine lt_of_not_ge fun h => ?_
cases' le.dest h with k e
injection s.symm.trans (psub_eq_some.2 <| (add_comm _ _).trans e)
· show n ≤ m
rw [← psub_eq_some.1 s]
apply Nat.le_add_left
| 149 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : ℕ → Option ℕ
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@[simp]
theorem ppred_zero : ppred 0 = none := rfl
@[simp]
theorem ppred_succ {n : ℕ} : ppred (succ n) = some n := rfl
def psub (m : ℕ) : ℕ → Option ℕ
| 0 => some m
| n + 1 => psub m n >>= ppred
#align nat.psub Nat.psub
@[simp]
theorem psub_zero {m : ℕ} : psub m 0 = some m := rfl
@[simp]
theorem psub_succ {m n : ℕ} : psub m (succ n) = psub m n >>= ppred := rfl
theorem pred_eq_ppred (n : ℕ) : pred n = (ppred n).getD 0 := by cases n <;> rfl
#align nat.pred_eq_ppred Nat.pred_eq_ppred
theorem sub_eq_psub (m : ℕ) : ∀ n, m - n = (psub m n).getD 0
| 0 => rfl
| n + 1 => (pred_eq_ppred (m - n)).trans <| by rw [sub_eq_psub m n, psub]; cases psub m n <;> rfl
#align nat.sub_eq_psub Nat.sub_eq_psub
@[simp]
theorem ppred_eq_some {m : ℕ} : ∀ {n}, ppred n = some m ↔ succ m = n
| 0 => by constructor <;> intro h <;> contradiction
| n + 1 => by constructor <;> intro h <;> injection h <;> subst m <;> rfl
#align nat.ppred_eq_some Nat.ppred_eq_some
-- Porting note: `contradiction` required an `intro` for the goals
-- `ppred (n + 1) = none → n + 1 = 0` and `n + 1 = 0 → ppred (n + 1) = none`
@[simp]
theorem ppred_eq_none : ∀ {n : ℕ}, ppred n = none ↔ n = 0
| 0 => by simp
| n + 1 => by constructor <;> intro <;> contradiction
#align nat.ppred_eq_none Nat.ppred_eq_none
theorem psub_eq_some {m : ℕ} : ∀ {n k}, psub m n = some k ↔ k + n = m
| 0, k => by simp [eq_comm]
| n + 1, k => by
apply Option.bind_eq_some.trans
simp only [psub_eq_some, ppred_eq_some]
simp [add_comm, add_left_comm, Nat.succ_eq_add_one]
#align nat.psub_eq_some Nat.psub_eq_some
theorem psub_eq_none {m n : ℕ} : psub m n = none ↔ m < n := by
cases s : psub m n <;> simp [eq_comm]
· show m < n
refine lt_of_not_ge fun h => ?_
cases' le.dest h with k e
injection s.symm.trans (psub_eq_some.2 <| (add_comm _ _).trans e)
· show n ≤ m
rw [← psub_eq_some.1 s]
apply Nat.le_add_left
#align nat.psub_eq_none Nat.psub_eq_none
theorem ppred_eq_pred {n} (h : 0 < n) : ppred n = some (pred n) :=
ppred_eq_some.2 <| succ_pred_eq_of_pos h
#align nat.ppred_eq_pred Nat.ppred_eq_pred
theorem psub_eq_sub {m n} (h : n ≤ m) : psub m n = some (m - n) :=
psub_eq_some.2 <| Nat.sub_add_cancel h
#align nat.psub_eq_sub Nat.psub_eq_sub
-- Porting note: we only have the simp lemma `Option.bind_some` which uses `Option.bind` not `>>=`
| Mathlib/Data/Nat/PSub.lean | 105 | 109 | theorem psub_add (m n k) :
psub m (n + k) = (do psub (← psub m n) k) := by |
induction k with
| zero => simp only [zero_eq, add_zero, psub_zero, Option.bind_eq_bind, Option.bind_some]
| succ n ih => simp only [ih, add_succ, psub_succ, bind_assoc]
| 149 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : ℕ → Option ℕ
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@[simp]
theorem ppred_zero : ppred 0 = none := rfl
@[simp]
theorem ppred_succ {n : ℕ} : ppred (succ n) = some n := rfl
def psub (m : ℕ) : ℕ → Option ℕ
| 0 => some m
| n + 1 => psub m n >>= ppred
#align nat.psub Nat.psub
@[simp]
theorem psub_zero {m : ℕ} : psub m 0 = some m := rfl
@[simp]
theorem psub_succ {m n : ℕ} : psub m (succ n) = psub m n >>= ppred := rfl
theorem pred_eq_ppred (n : ℕ) : pred n = (ppred n).getD 0 := by cases n <;> rfl
#align nat.pred_eq_ppred Nat.pred_eq_ppred
theorem sub_eq_psub (m : ℕ) : ∀ n, m - n = (psub m n).getD 0
| 0 => rfl
| n + 1 => (pred_eq_ppred (m - n)).trans <| by rw [sub_eq_psub m n, psub]; cases psub m n <;> rfl
#align nat.sub_eq_psub Nat.sub_eq_psub
@[simp]
theorem ppred_eq_some {m : ℕ} : ∀ {n}, ppred n = some m ↔ succ m = n
| 0 => by constructor <;> intro h <;> contradiction
| n + 1 => by constructor <;> intro h <;> injection h <;> subst m <;> rfl
#align nat.ppred_eq_some Nat.ppred_eq_some
-- Porting note: `contradiction` required an `intro` for the goals
-- `ppred (n + 1) = none → n + 1 = 0` and `n + 1 = 0 → ppred (n + 1) = none`
@[simp]
theorem ppred_eq_none : ∀ {n : ℕ}, ppred n = none ↔ n = 0
| 0 => by simp
| n + 1 => by constructor <;> intro <;> contradiction
#align nat.ppred_eq_none Nat.ppred_eq_none
theorem psub_eq_some {m : ℕ} : ∀ {n k}, psub m n = some k ↔ k + n = m
| 0, k => by simp [eq_comm]
| n + 1, k => by
apply Option.bind_eq_some.trans
simp only [psub_eq_some, ppred_eq_some]
simp [add_comm, add_left_comm, Nat.succ_eq_add_one]
#align nat.psub_eq_some Nat.psub_eq_some
theorem psub_eq_none {m n : ℕ} : psub m n = none ↔ m < n := by
cases s : psub m n <;> simp [eq_comm]
· show m < n
refine lt_of_not_ge fun h => ?_
cases' le.dest h with k e
injection s.symm.trans (psub_eq_some.2 <| (add_comm _ _).trans e)
· show n ≤ m
rw [← psub_eq_some.1 s]
apply Nat.le_add_left
#align nat.psub_eq_none Nat.psub_eq_none
theorem ppred_eq_pred {n} (h : 0 < n) : ppred n = some (pred n) :=
ppred_eq_some.2 <| succ_pred_eq_of_pos h
#align nat.ppred_eq_pred Nat.ppred_eq_pred
theorem psub_eq_sub {m n} (h : n ≤ m) : psub m n = some (m - n) :=
psub_eq_some.2 <| Nat.sub_add_cancel h
#align nat.psub_eq_sub Nat.psub_eq_sub
-- Porting note: we only have the simp lemma `Option.bind_some` which uses `Option.bind` not `>>=`
theorem psub_add (m n k) :
psub m (n + k) = (do psub (← psub m n) k) := by
induction k with
| zero => simp only [zero_eq, add_zero, psub_zero, Option.bind_eq_bind, Option.bind_some]
| succ n ih => simp only [ih, add_succ, psub_succ, bind_assoc]
#align nat.psub_add Nat.psub_add
@[inline]
def psub' (m n : ℕ) : Option ℕ :=
if n ≤ m then some (m - n) else none
#align nat.psub' Nat.psub'
| Mathlib/Data/Nat/PSub.lean | 118 | 122 | theorem psub'_eq_psub (m n) : psub' m n = psub m n := by |
rw [psub']
split_ifs with h
· exact (psub_eq_sub h).symm
· exact (psub_eq_none.2 (not_le.1 h)).symm
| 149 |
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
-- This lemma competes with `Int.ofNat_eq_natCast` to come later
@[simp high, nolint simpNF, norm_cast]
theorem cast_natCast (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_natCastₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_natCastₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
| Mathlib/Data/Int/Cast/Basic.lean | 74 | 76 | theorem cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by |
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
| 150 |
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
-- This lemma competes with `Int.ofNat_eq_natCast` to come later
@[simp high, nolint simpNF, norm_cast]
theorem cast_natCast (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_natCastₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_natCastₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
| Mathlib/Data/Int/Cast/Basic.lean | 79 | 80 | theorem cast_one : ((1 : ℤ) : R) = 1 := by |
erw [cast_natCast, Nat.cast_one]
| 150 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
| Mathlib/Data/Sum/Basic.lean | 27 | 30 | theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by |
rw [← not_forall_not, forall_sum]
simp
| 151 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum]
simp
theorem inl_injective : Function.Injective (inl : α → Sum α β) := fun _ _ ↦ inl.inj
#align sum.inl_injective Sum.inl_injective
theorem inr_injective : Function.Injective (inr : β → Sum α β) := fun _ _ ↦ inr.inj
#align sum.inr_injective Sum.inr_injective
| Mathlib/Data/Sum/Basic.lean | 38 | 40 | theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀ i, P (inr i))
{x y : α ⊕ β} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by | cases h; rfl
| 151 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum]
simp
theorem inl_injective : Function.Injective (inl : α → Sum α β) := fun _ _ ↦ inl.inj
#align sum.inl_injective Sum.inl_injective
theorem inr_injective : Function.Injective (inr : β → Sum α β) := fun _ _ ↦ inr.inj
#align sum.inr_injective Sum.inr_injective
theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀ i, P (inr i))
{x y : α ⊕ β} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by cases h; rfl
section get
#align sum.is_left Sum.isLeft
#align sum.is_right Sum.isRight
#align sum.get_left Sum.getLeft?
#align sum.get_right Sum.getRight?
variable {x y : Sum α β}
#align sum.get_left_eq_none_iff Sum.getLeft?_eq_none_iff
#align sum.get_right_eq_none_iff Sum.getRight?_eq_none_iff
| Mathlib/Data/Sum/Basic.lean | 54 | 55 | theorem eq_left_iff_getLeft_eq {a : α} : x = inl a ↔ ∃ h, x.getLeft h = a := by |
cases x <;> simp
| 151 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum]
simp
theorem inl_injective : Function.Injective (inl : α → Sum α β) := fun _ _ ↦ inl.inj
#align sum.inl_injective Sum.inl_injective
theorem inr_injective : Function.Injective (inr : β → Sum α β) := fun _ _ ↦ inr.inj
#align sum.inr_injective Sum.inr_injective
theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀ i, P (inr i))
{x y : α ⊕ β} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by cases h; rfl
section get
#align sum.is_left Sum.isLeft
#align sum.is_right Sum.isRight
#align sum.get_left Sum.getLeft?
#align sum.get_right Sum.getRight?
variable {x y : Sum α β}
#align sum.get_left_eq_none_iff Sum.getLeft?_eq_none_iff
#align sum.get_right_eq_none_iff Sum.getRight?_eq_none_iff
theorem eq_left_iff_getLeft_eq {a : α} : x = inl a ↔ ∃ h, x.getLeft h = a := by
cases x <;> simp
| Mathlib/Data/Sum/Basic.lean | 57 | 58 | theorem eq_right_iff_getRight_eq {b : β} : x = inr b ↔ ∃ h, x.getRight h = b := by |
cases x <;> simp
| 151 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum]
simp
theorem inl_injective : Function.Injective (inl : α → Sum α β) := fun _ _ ↦ inl.inj
#align sum.inl_injective Sum.inl_injective
theorem inr_injective : Function.Injective (inr : β → Sum α β) := fun _ _ ↦ inr.inj
#align sum.inr_injective Sum.inr_injective
theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀ i, P (inr i))
{x y : α ⊕ β} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by cases h; rfl
section get
#align sum.is_left Sum.isLeft
#align sum.is_right Sum.isRight
#align sum.get_left Sum.getLeft?
#align sum.get_right Sum.getRight?
variable {x y : Sum α β}
#align sum.get_left_eq_none_iff Sum.getLeft?_eq_none_iff
#align sum.get_right_eq_none_iff Sum.getRight?_eq_none_iff
theorem eq_left_iff_getLeft_eq {a : α} : x = inl a ↔ ∃ h, x.getLeft h = a := by
cases x <;> simp
theorem eq_right_iff_getRight_eq {b : β} : x = inr b ↔ ∃ h, x.getRight h = b := by
cases x <;> simp
#align sum.get_left_eq_some_iff Sum.getLeft?_eq_some_iff
#align sum.get_right_eq_some_iff Sum.getRight?_eq_some_iff
| Mathlib/Data/Sum/Basic.lean | 63 | 64 | theorem getLeft_eq_getLeft? (h₁ : x.isLeft) (h₂ : x.getLeft?.isSome) :
x.getLeft h₁ = x.getLeft?.get h₂ := by | simp [← getLeft?_eq_some_iff]
| 151 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum]
simp
theorem inl_injective : Function.Injective (inl : α → Sum α β) := fun _ _ ↦ inl.inj
#align sum.inl_injective Sum.inl_injective
theorem inr_injective : Function.Injective (inr : β → Sum α β) := fun _ _ ↦ inr.inj
#align sum.inr_injective Sum.inr_injective
theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀ i, P (inr i))
{x y : α ⊕ β} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by cases h; rfl
section get
#align sum.is_left Sum.isLeft
#align sum.is_right Sum.isRight
#align sum.get_left Sum.getLeft?
#align sum.get_right Sum.getRight?
variable {x y : Sum α β}
#align sum.get_left_eq_none_iff Sum.getLeft?_eq_none_iff
#align sum.get_right_eq_none_iff Sum.getRight?_eq_none_iff
theorem eq_left_iff_getLeft_eq {a : α} : x = inl a ↔ ∃ h, x.getLeft h = a := by
cases x <;> simp
theorem eq_right_iff_getRight_eq {b : β} : x = inr b ↔ ∃ h, x.getRight h = b := by
cases x <;> simp
#align sum.get_left_eq_some_iff Sum.getLeft?_eq_some_iff
#align sum.get_right_eq_some_iff Sum.getRight?_eq_some_iff
theorem getLeft_eq_getLeft? (h₁ : x.isLeft) (h₂ : x.getLeft?.isSome) :
x.getLeft h₁ = x.getLeft?.get h₂ := by simp [← getLeft?_eq_some_iff]
| Mathlib/Data/Sum/Basic.lean | 66 | 67 | theorem getRight_eq_getRight? (h₁ : x.isRight) (h₂ : x.getRight?.isSome) :
x.getRight h₁ = x.getRight?.get h₂ := by | simp [← getRight?_eq_some_iff]
| 151 |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.forall Sum.forall
#align sum.exists Sum.exists
theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum]
simp
theorem inl_injective : Function.Injective (inl : α → Sum α β) := fun _ _ ↦ inl.inj
#align sum.inl_injective Sum.inl_injective
theorem inr_injective : Function.Injective (inr : β → Sum α β) := fun _ _ ↦ inr.inj
#align sum.inr_injective Sum.inr_injective
theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀ i, P (inr i))
{x y : α ⊕ β} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by cases h; rfl
#align sum.inl.inj_iff Sum.inl.inj_iff
#align sum.inr.inj_iff Sum.inr.inj_iff
#align sum.inl_ne_inr Sum.inl_ne_inr
#align sum.inr_ne_inl Sum.inr_ne_inl
#align sum.elim Sum.elim
#align sum.elim_inl Sum.elim_inl
#align sum.elim_inr Sum.elim_inr
#align sum.elim_comp_inl Sum.elim_comp_inl
#align sum.elim_comp_inr Sum.elim_comp_inr
#align sum.elim_inl_inr Sum.elim_inl_inr
#align sum.comp_elim Sum.comp_elim
#align sum.elim_comp_inl_inr Sum.elim_comp_inl_inr
#align sum.map Sum.map
#align sum.map_inl Sum.map_inl
#align sum.map_inr Sum.map_inr
#align sum.map_map Sum.map_map
#align sum.map_comp_map Sum.map_comp_map
#align sum.map_id_id Sum.map_id_id
#align sum.elim_map Sum.elim_map
#align sum.elim_comp_map Sum.elim_comp_map
#align sum.is_left_map Sum.isLeft_map
#align sum.is_right_map Sum.isRight_map
#align sum.get_left_map Sum.getLeft?_map
#align sum.get_right_map Sum.getRight?_map
open Function (update update_eq_iff update_comp_eq_of_injective update_comp_eq_of_forall_ne)
@[simp]
theorem update_elim_inl [DecidableEq α] [DecidableEq (Sum α β)] {f : α → γ} {g : β → γ} {i : α}
{x : γ} : update (Sum.elim f g) (inl i) x = Sum.elim (update f i x) g :=
update_eq_iff.2 ⟨by simp, by simp (config := { contextual := true })⟩
#align sum.update_elim_inl Sum.update_elim_inl
@[simp]
theorem update_elim_inr [DecidableEq β] [DecidableEq (Sum α β)] {f : α → γ} {g : β → γ} {i : β}
{x : γ} : update (Sum.elim f g) (inr i) x = Sum.elim f (update g i x) :=
update_eq_iff.2 ⟨by simp, by simp (config := { contextual := true })⟩
#align sum.update_elim_inr Sum.update_elim_inr
@[simp]
theorem update_inl_comp_inl [DecidableEq α] [DecidableEq (Sum α β)] {f : Sum α β → γ} {i : α}
{x : γ} : update f (inl i) x ∘ inl = update (f ∘ inl) i x :=
update_comp_eq_of_injective _ inl_injective _ _
#align sum.update_inl_comp_inl Sum.update_inl_comp_inl
@[simp]
| Mathlib/Data/Sum/Basic.lean | 132 | 134 | theorem update_inl_apply_inl [DecidableEq α] [DecidableEq (Sum α β)] {f : Sum α β → γ} {i j : α}
{x : γ} : update f (inl i) x (inl j) = update (f ∘ inl) i x j := by |
rw [← update_inl_comp_inl, Function.comp_apply]
| 151 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
| Mathlib/Topology/Baire/Lemmas.lean | 50 | 55 | theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by |
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
| 152 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
#align dense_sInter_of_open dense_sInter_of_isOpen
| Mathlib/Topology/Baire/Lemmas.lean | 60 | 63 | theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by |
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
| 152 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
#align dense_sInter_of_open dense_sInter_of_isOpen
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
#align dense_bInter_of_open dense_biInter_of_isOpen
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
(hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)
#align dense_Inter_of_open dense_iInter_of_isOpen
| Mathlib/Topology/Baire/Lemmas.lean | 74 | 81 | theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by |
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
| 152 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
#align dense_sInter_of_open dense_sInter_of_isOpen
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
#align dense_bInter_of_open dense_biInter_of_isOpen
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
(hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)
#align dense_Inter_of_open dense_iInter_of_isOpen
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
#align mem_residual mem_residual
| Mathlib/Topology/Baire/Lemmas.lean | 85 | 88 | theorem eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by |
simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
tauto
| 152 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
#align dense_sInter_of_open dense_sInter_of_isOpen
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
#align dense_bInter_of_open dense_biInter_of_isOpen
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
(hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)
#align dense_Inter_of_open dense_iInter_of_isOpen
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
#align mem_residual mem_residual
theorem eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by
simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
tauto
#align eventually_residual eventually_residual
theorem dense_of_mem_residual {s : Set X} (hs : s ∈ residual X) : Dense s :=
let ⟨_, hts, _, hd⟩ := mem_residual.1 hs
hd.mono hts
#align dense_of_mem_residual dense_of_mem_residual
theorem dense_sInter_of_Gδ {S : Set (Set X)} (ho : ∀ s ∈ S, IsGδ s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) :=
dense_of_mem_residual ((countable_sInter_mem hS).mpr
(fun _ hs => residual_of_dense_Gδ (ho _ hs) (hd _ hs)))
set_option linter.uppercaseLean3 false in
#align dense_sInter_of_Gδ dense_sInter_of_Gδ
theorem dense_iInter_of_Gδ [Countable ι] {f : ι → Set X} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) :=
dense_sInter_of_Gδ (forall_mem_range.2 ‹_›) (countable_range _) (forall_mem_range.2 ‹_›)
set_option linter.uppercaseLean3 false in
#align dense_Inter_of_Gδ dense_iInter_of_Gδ
| Mathlib/Topology/Baire/Lemmas.lean | 114 | 118 | theorem dense_biInter_of_Gδ {S : Set α} {f : ∀ x ∈ S, Set X} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ S, f s ‹_›) := by |
rw [biInter_eq_iInter]
haveI := hS.to_subtype
exact dense_iInter_of_Gδ (fun s => ho s s.2) fun s => hd s s.2
| 152 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
#align dense_sInter_of_open dense_sInter_of_isOpen
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
#align dense_bInter_of_open dense_biInter_of_isOpen
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
(hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)
#align dense_Inter_of_open dense_iInter_of_isOpen
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
#align mem_residual mem_residual
theorem eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by
simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
tauto
#align eventually_residual eventually_residual
theorem dense_of_mem_residual {s : Set X} (hs : s ∈ residual X) : Dense s :=
let ⟨_, hts, _, hd⟩ := mem_residual.1 hs
hd.mono hts
#align dense_of_mem_residual dense_of_mem_residual
theorem dense_sInter_of_Gδ {S : Set (Set X)} (ho : ∀ s ∈ S, IsGδ s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) :=
dense_of_mem_residual ((countable_sInter_mem hS).mpr
(fun _ hs => residual_of_dense_Gδ (ho _ hs) (hd _ hs)))
set_option linter.uppercaseLean3 false in
#align dense_sInter_of_Gδ dense_sInter_of_Gδ
theorem dense_iInter_of_Gδ [Countable ι] {f : ι → Set X} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) :=
dense_sInter_of_Gδ (forall_mem_range.2 ‹_›) (countable_range _) (forall_mem_range.2 ‹_›)
set_option linter.uppercaseLean3 false in
#align dense_Inter_of_Gδ dense_iInter_of_Gδ
theorem dense_biInter_of_Gδ {S : Set α} {f : ∀ x ∈ S, Set X} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ S, f s ‹_›) := by
rw [biInter_eq_iInter]
haveI := hS.to_subtype
exact dense_iInter_of_Gδ (fun s => ho s s.2) fun s => hd s s.2
set_option linter.uppercaseLean3 false in
#align dense_bInter_of_Gδ dense_biInter_of_Gδ
| Mathlib/Topology/Baire/Lemmas.lean | 123 | 126 | theorem Dense.inter_of_Gδ {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by |
rw [inter_eq_iInter]
apply dense_iInter_of_Gδ <;> simp [Bool.forall_bool, *]
| 152 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace X] [BaireSpace X]
theorem dense_iInter_of_isOpen_nat {f : ℕ → Set X} (ho : ∀ n, IsOpen (f n))
(hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) :=
BaireSpace.baire_property f ho hd
#align dense_Inter_of_open_nat dense_iInter_of_isOpen_nat
theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
#align dense_sInter_of_open dense_sInter_of_isOpen
theorem dense_biInter_of_isOpen {S : Set α} {f : α → Set X} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
rw [← sInter_image]
refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image]
#align dense_bInter_of_open dense_biInter_of_isOpen
theorem dense_iInter_of_isOpen [Countable ι] {f : ι → Set X} (ho : ∀ i, IsOpen (f i))
(hd : ∀ i, Dense (f i)) : Dense (⋂ s, f s) :=
dense_sInter_of_isOpen (forall_mem_range.2 ho) (countable_range _) (forall_mem_range.2 hd)
#align dense_Inter_of_open dense_iInter_of_isOpen
theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
#align mem_residual mem_residual
theorem eventually_residual {p : X → Prop} :
(∀ᶠ x in residual X, p x) ↔ ∃ t : Set X, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x := by
simp only [Filter.Eventually, mem_residual, subset_def, mem_setOf_eq]
tauto
#align eventually_residual eventually_residual
theorem dense_of_mem_residual {s : Set X} (hs : s ∈ residual X) : Dense s :=
let ⟨_, hts, _, hd⟩ := mem_residual.1 hs
hd.mono hts
#align dense_of_mem_residual dense_of_mem_residual
theorem dense_sInter_of_Gδ {S : Set (Set X)} (ho : ∀ s ∈ S, IsGδ s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) :=
dense_of_mem_residual ((countable_sInter_mem hS).mpr
(fun _ hs => residual_of_dense_Gδ (ho _ hs) (hd _ hs)))
set_option linter.uppercaseLean3 false in
#align dense_sInter_of_Gδ dense_sInter_of_Gδ
theorem dense_iInter_of_Gδ [Countable ι] {f : ι → Set X} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) :=
dense_sInter_of_Gδ (forall_mem_range.2 ‹_›) (countable_range _) (forall_mem_range.2 ‹_›)
set_option linter.uppercaseLean3 false in
#align dense_Inter_of_Gδ dense_iInter_of_Gδ
theorem dense_biInter_of_Gδ {S : Set α} {f : ∀ x ∈ S, Set X} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ S, f s ‹_›) := by
rw [biInter_eq_iInter]
haveI := hS.to_subtype
exact dense_iInter_of_Gδ (fun s => ho s s.2) fun s => hd s s.2
set_option linter.uppercaseLean3 false in
#align dense_bInter_of_Gδ dense_biInter_of_Gδ
theorem Dense.inter_of_Gδ {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by
rw [inter_eq_iInter]
apply dense_iInter_of_Gδ <;> simp [Bool.forall_bool, *]
set_option linter.uppercaseLean3 false in
#align dense.inter_of_Gδ Dense.inter_of_Gδ
| Mathlib/Topology/Baire/Lemmas.lean | 132 | 145 | theorem IsGδ.dense_iUnion_interior_of_closed [Countable ι] {s : Set X} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set X} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ i, interior (f i)) := by |
let g i := (frontier (f i))ᶜ
have hgo : ∀ i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl
have hgd : Dense (⋂ i, g i) := by
refine dense_iInter_of_isOpen hgo fun i x => ?_
rw [closure_compl, interior_frontier (hc _)]
exact id
refine (hd.inter_of_Gδ hs (.iInter_of_isOpen fun i => (hgo i)) hgd).mono ?_
rintro x ⟨hxs, hxg⟩
rw [mem_iInter] at hxg
rcases mem_iUnion.1 (hU hxs) with ⟨i, hi⟩
exact mem_iUnion.2 ⟨i, self_diff_frontier (f i) ▸ ⟨hi, hxg _⟩⟩
| 152 |
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca092c54b8bf04f0f10592f59489"
variable {l m n o p q : Type*} {m' n' p' : o → Type*}
variable {R : Type*} {S : Type*} {α : Type*} {β : Type*}
open Matrix
namespace Matrix
theorem dotProduct_block [Fintype m] [Fintype n] [Mul α] [AddCommMonoid α] (v w : Sum m n → α) :
v ⬝ᵥ w = v ∘ Sum.inl ⬝ᵥ w ∘ Sum.inl + v ∘ Sum.inr ⬝ᵥ w ∘ Sum.inr :=
Fintype.sum_sum_type _
#align matrix.dot_product_block Matrix.dotProduct_block
section BlockMatrices
-- @[pp_nodot] -- Porting note: removed
def fromBlocks (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α) (D : Matrix o m α) :
Matrix (Sum n o) (Sum l m) α :=
of <| Sum.elim (fun i => Sum.elim (A i) (B i)) fun i => Sum.elim (C i) (D i)
#align matrix.from_blocks Matrix.fromBlocks
@[simp]
theorem fromBlocks_apply₁₁ (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (i : n) (j : l) : fromBlocks A B C D (Sum.inl i) (Sum.inl j) = A i j :=
rfl
#align matrix.from_blocks_apply₁₁ Matrix.fromBlocks_apply₁₁
@[simp]
theorem fromBlocks_apply₁₂ (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (i : n) (j : m) : fromBlocks A B C D (Sum.inl i) (Sum.inr j) = B i j :=
rfl
#align matrix.from_blocks_apply₁₂ Matrix.fromBlocks_apply₁₂
@[simp]
theorem fromBlocks_apply₂₁ (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (i : o) (j : l) : fromBlocks A B C D (Sum.inr i) (Sum.inl j) = C i j :=
rfl
#align matrix.from_blocks_apply₂₁ Matrix.fromBlocks_apply₂₁
@[simp]
theorem fromBlocks_apply₂₂ (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (i : o) (j : m) : fromBlocks A B C D (Sum.inr i) (Sum.inr j) = D i j :=
rfl
#align matrix.from_blocks_apply₂₂ Matrix.fromBlocks_apply₂₂
def toBlocks₁₁ (M : Matrix (Sum n o) (Sum l m) α) : Matrix n l α :=
of fun i j => M (Sum.inl i) (Sum.inl j)
#align matrix.to_blocks₁₁ Matrix.toBlocks₁₁
def toBlocks₁₂ (M : Matrix (Sum n o) (Sum l m) α) : Matrix n m α :=
of fun i j => M (Sum.inl i) (Sum.inr j)
#align matrix.to_blocks₁₂ Matrix.toBlocks₁₂
def toBlocks₂₁ (M : Matrix (Sum n o) (Sum l m) α) : Matrix o l α :=
of fun i j => M (Sum.inr i) (Sum.inl j)
#align matrix.to_blocks₂₁ Matrix.toBlocks₂₁
def toBlocks₂₂ (M : Matrix (Sum n o) (Sum l m) α) : Matrix o m α :=
of fun i j => M (Sum.inr i) (Sum.inr j)
#align matrix.to_blocks₂₂ Matrix.toBlocks₂₂
| Mathlib/Data/Matrix/Block.lean | 97 | 100 | theorem fromBlocks_toBlocks (M : Matrix (Sum n o) (Sum l m) α) :
fromBlocks M.toBlocks₁₁ M.toBlocks₁₂ M.toBlocks₂₁ M.toBlocks₂₂ = M := by |
ext i j
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl
| 153 |
import Mathlib.Data.Matrix.Block
#align_import linear_algebra.matrix.symmetric from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β n m R : Type*}
namespace Matrix
open Matrix
def IsSymm (A : Matrix n n α) : Prop :=
Aᵀ = A
#align matrix.is_symm Matrix.IsSymm
instance (A : Matrix n n α) [Decidable (Aᵀ = A)] : Decidable (IsSymm A) :=
inferInstanceAs <| Decidable (_ = _)
theorem IsSymm.eq {A : Matrix n n α} (h : A.IsSymm) : Aᵀ = A :=
h
#align matrix.is_symm.eq Matrix.IsSymm.eq
theorem IsSymm.ext_iff {A : Matrix n n α} : A.IsSymm ↔ ∀ i j, A j i = A i j :=
Matrix.ext_iff.symm
#align matrix.is_symm.ext_iff Matrix.IsSymm.ext_iff
-- @[ext] -- Porting note: removed attribute
theorem IsSymm.ext {A : Matrix n n α} : (∀ i j, A j i = A i j) → A.IsSymm :=
Matrix.ext
#align matrix.is_symm.ext Matrix.IsSymm.ext
theorem IsSymm.apply {A : Matrix n n α} (h : A.IsSymm) (i j : n) : A j i = A i j :=
IsSymm.ext_iff.1 h i j
#align matrix.is_symm.apply Matrix.IsSymm.apply
theorem isSymm_mul_transpose_self [Fintype n] [CommSemiring α] (A : Matrix n n α) :
(A * Aᵀ).IsSymm :=
transpose_mul _ _
#align matrix.is_symm_mul_transpose_self Matrix.isSymm_mul_transpose_self
theorem isSymm_transpose_mul_self [Fintype n] [CommSemiring α] (A : Matrix n n α) :
(Aᵀ * A).IsSymm :=
transpose_mul _ _
#align matrix.is_symm_transpose_mul_self Matrix.isSymm_transpose_mul_self
theorem isSymm_add_transpose_self [AddCommSemigroup α] (A : Matrix n n α) : (A + Aᵀ).IsSymm :=
add_comm _ _
#align matrix.is_symm_add_transpose_self Matrix.isSymm_add_transpose_self
theorem isSymm_transpose_add_self [AddCommSemigroup α] (A : Matrix n n α) : (Aᵀ + A).IsSymm :=
add_comm _ _
#align matrix.is_symm_transpose_add_self Matrix.isSymm_transpose_add_self
@[simp]
theorem isSymm_zero [Zero α] : (0 : Matrix n n α).IsSymm :=
transpose_zero
#align matrix.is_symm_zero Matrix.isSymm_zero
@[simp]
theorem isSymm_one [DecidableEq n] [Zero α] [One α] : (1 : Matrix n n α).IsSymm :=
transpose_one
#align matrix.is_symm_one Matrix.isSymm_one
| Mathlib/LinearAlgebra/Matrix/Symmetric.lean | 86 | 89 | theorem IsSymm.pow [CommSemiring α] [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsSymm)
(k : ℕ) :
(A ^ k).IsSymm := by |
rw [IsSymm, transpose_pow, h]
| 154 |
import Mathlib.Data.Matrix.Block
#align_import linear_algebra.matrix.symmetric from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β n m R : Type*}
namespace Matrix
open Matrix
def IsSymm (A : Matrix n n α) : Prop :=
Aᵀ = A
#align matrix.is_symm Matrix.IsSymm
instance (A : Matrix n n α) [Decidable (Aᵀ = A)] : Decidable (IsSymm A) :=
inferInstanceAs <| Decidable (_ = _)
theorem IsSymm.eq {A : Matrix n n α} (h : A.IsSymm) : Aᵀ = A :=
h
#align matrix.is_symm.eq Matrix.IsSymm.eq
theorem IsSymm.ext_iff {A : Matrix n n α} : A.IsSymm ↔ ∀ i j, A j i = A i j :=
Matrix.ext_iff.symm
#align matrix.is_symm.ext_iff Matrix.IsSymm.ext_iff
-- @[ext] -- Porting note: removed attribute
theorem IsSymm.ext {A : Matrix n n α} : (∀ i j, A j i = A i j) → A.IsSymm :=
Matrix.ext
#align matrix.is_symm.ext Matrix.IsSymm.ext
theorem IsSymm.apply {A : Matrix n n α} (h : A.IsSymm) (i j : n) : A j i = A i j :=
IsSymm.ext_iff.1 h i j
#align matrix.is_symm.apply Matrix.IsSymm.apply
theorem isSymm_mul_transpose_self [Fintype n] [CommSemiring α] (A : Matrix n n α) :
(A * Aᵀ).IsSymm :=
transpose_mul _ _
#align matrix.is_symm_mul_transpose_self Matrix.isSymm_mul_transpose_self
theorem isSymm_transpose_mul_self [Fintype n] [CommSemiring α] (A : Matrix n n α) :
(Aᵀ * A).IsSymm :=
transpose_mul _ _
#align matrix.is_symm_transpose_mul_self Matrix.isSymm_transpose_mul_self
theorem isSymm_add_transpose_self [AddCommSemigroup α] (A : Matrix n n α) : (A + Aᵀ).IsSymm :=
add_comm _ _
#align matrix.is_symm_add_transpose_self Matrix.isSymm_add_transpose_self
theorem isSymm_transpose_add_self [AddCommSemigroup α] (A : Matrix n n α) : (Aᵀ + A).IsSymm :=
add_comm _ _
#align matrix.is_symm_transpose_add_self Matrix.isSymm_transpose_add_self
@[simp]
theorem isSymm_zero [Zero α] : (0 : Matrix n n α).IsSymm :=
transpose_zero
#align matrix.is_symm_zero Matrix.isSymm_zero
@[simp]
theorem isSymm_one [DecidableEq n] [Zero α] [One α] : (1 : Matrix n n α).IsSymm :=
transpose_one
#align matrix.is_symm_one Matrix.isSymm_one
theorem IsSymm.pow [CommSemiring α] [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsSymm)
(k : ℕ) :
(A ^ k).IsSymm := by
rw [IsSymm, transpose_pow, h]
@[simp]
theorem IsSymm.map {A : Matrix n n α} (h : A.IsSymm) (f : α → β) : (A.map f).IsSymm :=
transpose_map.symm.trans (h.symm ▸ rfl)
#align matrix.is_symm.map Matrix.IsSymm.map
@[simp]
theorem IsSymm.transpose {A : Matrix n n α} (h : A.IsSymm) : Aᵀ.IsSymm :=
congr_arg _ h
#align matrix.is_symm.transpose Matrix.IsSymm.transpose
@[simp]
theorem IsSymm.conjTranspose [Star α] {A : Matrix n n α} (h : A.IsSymm) : Aᴴ.IsSymm :=
h.transpose.map _
#align matrix.is_symm.conj_transpose Matrix.IsSymm.conjTranspose
@[simp]
theorem IsSymm.neg [Neg α] {A : Matrix n n α} (h : A.IsSymm) : (-A).IsSymm :=
(transpose_neg _).trans (congr_arg _ h)
#align matrix.is_symm.neg Matrix.IsSymm.neg
@[simp]
theorem IsSymm.add {A B : Matrix n n α} [Add α] (hA : A.IsSymm) (hB : B.IsSymm) : (A + B).IsSymm :=
(transpose_add _ _).trans (hA.symm ▸ hB.symm ▸ rfl)
#align matrix.is_symm.add Matrix.IsSymm.add
@[simp]
theorem IsSymm.sub {A B : Matrix n n α} [Sub α] (hA : A.IsSymm) (hB : B.IsSymm) : (A - B).IsSymm :=
(transpose_sub _ _).trans (hA.symm ▸ hB.symm ▸ rfl)
#align matrix.is_symm.sub Matrix.IsSymm.sub
@[simp]
theorem IsSymm.smul [SMul R α] {A : Matrix n n α} (h : A.IsSymm) (k : R) : (k • A).IsSymm :=
(transpose_smul _ _).trans (congr_arg _ h)
#align matrix.is_symm.smul Matrix.IsSymm.smul
@[simp]
theorem IsSymm.submatrix {A : Matrix n n α} (h : A.IsSymm) (f : m → n) : (A.submatrix f f).IsSymm :=
(transpose_submatrix _ _ _).trans (h.symm ▸ rfl)
#align matrix.is_symm.submatrix Matrix.IsSymm.submatrix
@[simp]
theorem isSymm_diagonal [DecidableEq n] [Zero α] (v : n → α) : (diagonal v).IsSymm :=
diagonal_transpose _
#align matrix.is_symm_diagonal Matrix.isSymm_diagonal
| Mathlib/LinearAlgebra/Matrix/Symmetric.lean | 139 | 146 | theorem IsSymm.fromBlocks {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α}
{D : Matrix n n α} (hA : A.IsSymm) (hBC : Bᵀ = C) (hD : D.IsSymm) :
(A.fromBlocks B C D).IsSymm := by |
have hCB : Cᵀ = B := by
rw [← hBC]
simp
unfold Matrix.IsSymm
rw [fromBlocks_transpose, hA, hCB, hBC, hD]
| 154 |
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β γ m n : Type*}
variable {R : Type*}
namespace Matrix
open Matrix
def hadamard [Mul α] (A : Matrix m n α) (B : Matrix m n α) : Matrix m n α :=
of fun i j => A i j * B i j
#align matrix.hadamard Matrix.hadamard
-- TODO: set as an equation lemma for `hadamard`, see mathlib4#3024
@[simp]
theorem hadamard_apply [Mul α] (A : Matrix m n α) (B : Matrix m n α) (i j) :
hadamard A B i j = A i j * B i j :=
rfl
#align matrix.hadamard_apply Matrix.hadamard_apply
scoped infixl:100 " ⊙ " => Matrix.hadamard
section BasicProperties
variable (A : Matrix m n α) (B : Matrix m n α) (C : Matrix m n α)
-- commutativity
theorem hadamard_comm [CommSemigroup α] : A ⊙ B = B ⊙ A :=
ext fun _ _ => mul_comm _ _
#align matrix.hadamard_comm Matrix.hadamard_comm
-- associativity
theorem hadamard_assoc [Semigroup α] : A ⊙ B ⊙ C = A ⊙ (B ⊙ C) :=
ext fun _ _ => mul_assoc _ _ _
#align matrix.hadamard_assoc Matrix.hadamard_assoc
-- distributivity
theorem hadamard_add [Distrib α] : A ⊙ (B + C) = A ⊙ B + A ⊙ C :=
ext fun _ _ => left_distrib _ _ _
#align matrix.hadamard_add Matrix.hadamard_add
theorem add_hadamard [Distrib α] : (B + C) ⊙ A = B ⊙ A + C ⊙ A :=
ext fun _ _ => right_distrib _ _ _
#align matrix.add_hadamard Matrix.add_hadamard
-- scalar multiplication
section One
variable [DecidableEq n] [MulZeroOneClass α]
variable (M : Matrix n n α)
| Mathlib/Data/Matrix/Hadamard.lean | 116 | 118 | theorem hadamard_one : M ⊙ (1 : Matrix n n α) = diagonal fun i => M i i := by |
ext i j
by_cases h: i = j <;> simp [h]
| 155 |
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β γ m n : Type*}
variable {R : Type*}
namespace Matrix
open Matrix
def hadamard [Mul α] (A : Matrix m n α) (B : Matrix m n α) : Matrix m n α :=
of fun i j => A i j * B i j
#align matrix.hadamard Matrix.hadamard
-- TODO: set as an equation lemma for `hadamard`, see mathlib4#3024
@[simp]
theorem hadamard_apply [Mul α] (A : Matrix m n α) (B : Matrix m n α) (i j) :
hadamard A B i j = A i j * B i j :=
rfl
#align matrix.hadamard_apply Matrix.hadamard_apply
scoped infixl:100 " ⊙ " => Matrix.hadamard
section BasicProperties
variable (A : Matrix m n α) (B : Matrix m n α) (C : Matrix m n α)
-- commutativity
theorem hadamard_comm [CommSemigroup α] : A ⊙ B = B ⊙ A :=
ext fun _ _ => mul_comm _ _
#align matrix.hadamard_comm Matrix.hadamard_comm
-- associativity
theorem hadamard_assoc [Semigroup α] : A ⊙ B ⊙ C = A ⊙ (B ⊙ C) :=
ext fun _ _ => mul_assoc _ _ _
#align matrix.hadamard_assoc Matrix.hadamard_assoc
-- distributivity
theorem hadamard_add [Distrib α] : A ⊙ (B + C) = A ⊙ B + A ⊙ C :=
ext fun _ _ => left_distrib _ _ _
#align matrix.hadamard_add Matrix.hadamard_add
theorem add_hadamard [Distrib α] : (B + C) ⊙ A = B ⊙ A + C ⊙ A :=
ext fun _ _ => right_distrib _ _ _
#align matrix.add_hadamard Matrix.add_hadamard
-- scalar multiplication
section One
variable [DecidableEq n] [MulZeroOneClass α]
variable (M : Matrix n n α)
theorem hadamard_one : M ⊙ (1 : Matrix n n α) = diagonal fun i => M i i := by
ext i j
by_cases h: i = j <;> simp [h]
#align matrix.hadamard_one Matrix.hadamard_one
| Mathlib/Data/Matrix/Hadamard.lean | 121 | 123 | theorem one_hadamard : (1 : Matrix n n α) ⊙ M = diagonal fun i => M i i := by |
ext i j
by_cases h : i = j <;> simp [h]
| 155 |
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β γ m n : Type*}
variable {R : Type*}
namespace Matrix
open Matrix
def hadamard [Mul α] (A : Matrix m n α) (B : Matrix m n α) : Matrix m n α :=
of fun i j => A i j * B i j
#align matrix.hadamard Matrix.hadamard
-- TODO: set as an equation lemma for `hadamard`, see mathlib4#3024
@[simp]
theorem hadamard_apply [Mul α] (A : Matrix m n α) (B : Matrix m n α) (i j) :
hadamard A B i j = A i j * B i j :=
rfl
#align matrix.hadamard_apply Matrix.hadamard_apply
scoped infixl:100 " ⊙ " => Matrix.hadamard
section BasicProperties
variable (A : Matrix m n α) (B : Matrix m n α) (C : Matrix m n α)
-- commutativity
theorem hadamard_comm [CommSemigroup α] : A ⊙ B = B ⊙ A :=
ext fun _ _ => mul_comm _ _
#align matrix.hadamard_comm Matrix.hadamard_comm
-- associativity
theorem hadamard_assoc [Semigroup α] : A ⊙ B ⊙ C = A ⊙ (B ⊙ C) :=
ext fun _ _ => mul_assoc _ _ _
#align matrix.hadamard_assoc Matrix.hadamard_assoc
-- distributivity
theorem hadamard_add [Distrib α] : A ⊙ (B + C) = A ⊙ B + A ⊙ C :=
ext fun _ _ => left_distrib _ _ _
#align matrix.hadamard_add Matrix.hadamard_add
theorem add_hadamard [Distrib α] : (B + C) ⊙ A = B ⊙ A + C ⊙ A :=
ext fun _ _ => right_distrib _ _ _
#align matrix.add_hadamard Matrix.add_hadamard
-- scalar multiplication
section trace
variable [Fintype m] [Fintype n]
variable (R) [Semiring α] [Semiring R] [Module R α]
theorem sum_hadamard_eq : (∑ i : m, ∑ j : n, (A ⊙ B) i j) = trace (A * Bᵀ) :=
rfl
#align matrix.sum_hadamard_eq Matrix.sum_hadamard_eq
| Mathlib/Data/Matrix/Hadamard.lean | 148 | 151 | theorem dotProduct_vecMul_hadamard [DecidableEq m] [DecidableEq n] (v : m → α) (w : n → α) :
dotProduct (v ᵥ* (A ⊙ B)) w = trace (diagonal v * A * (B * diagonal w)ᵀ) := by |
rw [← sum_hadamard_eq, Finset.sum_comm]
simp [dotProduct, vecMul, Finset.sum_mul, mul_assoc]
| 155 |
import Mathlib.Data.DFinsupp.Order
#align_import data.dfinsupp.multiset from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
open Function
variable {α : Type*} {β : α → Type*}
namespace Multiset
variable [DecidableEq α] {s t : Multiset α}
def toDFinsupp : Multiset α →+ Π₀ _ : α, ℕ where
toFun s :=
{ toFun := fun n ↦ s.count n
support' := Trunc.mk ⟨s, fun i ↦ (em (i ∈ s)).imp_right Multiset.count_eq_zero_of_not_mem⟩ }
map_zero' := rfl
map_add' _ _ := DFinsupp.ext fun _ ↦ Multiset.count_add _ _ _
#align multiset.to_dfinsupp Multiset.toDFinsupp
@[simp]
theorem toDFinsupp_apply (s : Multiset α) (a : α) : Multiset.toDFinsupp s a = s.count a :=
rfl
#align multiset.to_dfinsupp_apply Multiset.toDFinsupp_apply
@[simp]
theorem toDFinsupp_support (s : Multiset α) : s.toDFinsupp.support = s.toFinset :=
Finset.filter_true_of_mem fun _ hx ↦ count_ne_zero.mpr <| Multiset.mem_toFinset.1 hx
#align multiset.to_dfinsupp_support Multiset.toDFinsupp_support
@[simp]
| Mathlib/Data/DFinsupp/Multiset.lean | 67 | 71 | theorem toDFinsupp_replicate (a : α) (n : ℕ) :
toDFinsupp (Multiset.replicate n a) = DFinsupp.single a n := by |
ext i
dsimp [toDFinsupp]
simp [count_replicate, eq_comm]
| 156 |
import Mathlib.Data.DFinsupp.Order
#align_import data.dfinsupp.multiset from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
open Function
variable {α : Type*} {β : α → Type*}
namespace Multiset
variable [DecidableEq α] {s t : Multiset α}
def toDFinsupp : Multiset α →+ Π₀ _ : α, ℕ where
toFun s :=
{ toFun := fun n ↦ s.count n
support' := Trunc.mk ⟨s, fun i ↦ (em (i ∈ s)).imp_right Multiset.count_eq_zero_of_not_mem⟩ }
map_zero' := rfl
map_add' _ _ := DFinsupp.ext fun _ ↦ Multiset.count_add _ _ _
#align multiset.to_dfinsupp Multiset.toDFinsupp
@[simp]
theorem toDFinsupp_apply (s : Multiset α) (a : α) : Multiset.toDFinsupp s a = s.count a :=
rfl
#align multiset.to_dfinsupp_apply Multiset.toDFinsupp_apply
@[simp]
theorem toDFinsupp_support (s : Multiset α) : s.toDFinsupp.support = s.toFinset :=
Finset.filter_true_of_mem fun _ hx ↦ count_ne_zero.mpr <| Multiset.mem_toFinset.1 hx
#align multiset.to_dfinsupp_support Multiset.toDFinsupp_support
@[simp]
theorem toDFinsupp_replicate (a : α) (n : ℕ) :
toDFinsupp (Multiset.replicate n a) = DFinsupp.single a n := by
ext i
dsimp [toDFinsupp]
simp [count_replicate, eq_comm]
#align multiset.to_dfinsupp_replicate Multiset.toDFinsupp_replicate
@[simp]
| Mathlib/Data/DFinsupp/Multiset.lean | 75 | 76 | theorem toDFinsupp_singleton (a : α) : toDFinsupp {a} = DFinsupp.single a 1 := by |
rw [← replicate_one, toDFinsupp_replicate]
| 156 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) : ∀ _ _ : Σ i, α i, Prop
| left {i j : ι} (a : α i) (b : α j) : r i j → Lex r s ⟨i, a⟩ ⟨j, b⟩
| right {i : ι} (a b : α i) : s i a b → Lex r s ⟨i, a⟩ ⟨i, b⟩
#align sigma.lex Sigma.Lex
| Mathlib/Data/Sigma/Lex.lean | 45 | 55 | theorem lex_iff : Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by |
constructor
· rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ _ h
| 157 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) : ∀ _ _ : Σ i, α i, Prop
| left {i j : ι} (a : α i) (b : α j) : r i j → Lex r s ⟨i, a⟩ ⟨j, b⟩
| right {i : ι} (a b : α i) : s i a b → Lex r s ⟨i, a⟩ ⟨i, b⟩
#align sigma.lex Sigma.Lex
theorem lex_iff : Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by
constructor
· rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ _ h
#align sigma.lex_iff Sigma.lex_iff
instance Lex.decidable (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) [DecidableEq ι]
[DecidableRel r] [∀ i, DecidableRel (s i)] : DecidableRel (Lex r s) := fun _ _ =>
decidable_of_decidable_of_iff lex_iff.symm
#align sigma.lex.decidable Sigma.Lex.decidable
| Mathlib/Data/Sigma/Lex.lean | 63 | 67 | theorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i}
(h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b := by |
obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h
· exact Lex.left _ _ (hr _ _ hij)
· exact Lex.right _ _ (hs _ _ _ hab)
| 157 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) : ∀ _ _ : Σ i, α i, Prop
| left {i j : ι} (a : α i) (b : α j) : r i j → Lex r s ⟨i, a⟩ ⟨j, b⟩
| right {i : ι} (a b : α i) : s i a b → Lex r s ⟨i, a⟩ ⟨i, b⟩
#align sigma.lex Sigma.Lex
theorem lex_iff : Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by
constructor
· rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ _ h
#align sigma.lex_iff Sigma.lex_iff
instance Lex.decidable (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) [DecidableEq ι]
[DecidableRel r] [∀ i, DecidableRel (s i)] : DecidableRel (Lex r s) := fun _ _ =>
decidable_of_decidable_of_iff lex_iff.symm
#align sigma.lex.decidable Sigma.Lex.decidable
theorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i}
(h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b := by
obtain ⟨a, b, hij⟩ | ⟨a, b, hab⟩ := h
· exact Lex.left _ _ (hr _ _ hij)
· exact Lex.right _ _ (hs _ _ _ hab)
#align sigma.lex.mono Sigma.Lex.mono
theorem Lex.mono_left (hr : ∀ a b, r₁ a b → r₂ a b) {a b : Σ i, α i} (h : Lex r₁ s a b) :
Lex r₂ s a b :=
h.mono hr fun _ _ _ => id
#align sigma.lex.mono_left Sigma.Lex.mono_left
theorem Lex.mono_right (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i} (h : Lex r s₁ a b) :
Lex r s₂ a b :=
h.mono (fun _ _ => id) hs
#align sigma.lex.mono_right Sigma.Lex.mono_right
| Mathlib/Data/Sigma/Lex.lean | 80 | 83 | theorem lex_swap : Lex (Function.swap r) s a b ↔ Lex r (fun i => Function.swap (s i)) b a := by |
constructor <;>
· rintro (⟨a, b, h⟩ | ⟨a, b, h⟩)
exacts [Lex.left _ _ h, Lex.right _ _ h]
| 157 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace PSigma
variable {ι : Sort*} {α : ι → Sort*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
| Mathlib/Data/Sigma/Lex.lean | 151 | 162 | theorem lex_iff {a b : Σ' i, α i} :
Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by |
constructor
· rintro (⟨a, b, hij⟩ | ⟨i, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ h
| 157 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace PSigma
variable {ι : Sort*} {α : ι → Sort*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
theorem lex_iff {a b : Σ' i, α i} :
Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by
constructor
· rintro (⟨a, b, hij⟩ | ⟨i, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ h
#align psigma.lex_iff PSigma.lex_iff
instance Lex.decidable (r : ι → ι → Prop) (s : ∀ i, α i → α i → Prop) [DecidableEq ι]
[DecidableRel r] [∀ i, DecidableRel (s i)] : DecidableRel (Lex r s) := fun _ _ =>
decidable_of_decidable_of_iff lex_iff.symm
#align psigma.lex.decidable PSigma.Lex.decidable
| Mathlib/Data/Sigma/Lex.lean | 170 | 175 | theorem Lex.mono {r₁ r₂ : ι → ι → Prop} {s₁ s₂ : ∀ i, α i → α i → Prop}
(hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ' i, α i}
(h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b := by |
obtain ⟨a, b, hij⟩ | ⟨i, hab⟩ := h
· exact Lex.left _ _ (hr _ _ hij)
· exact Lex.right _ (hs _ _ _ hab)
| 157 |
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.Scheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u
noncomputable section
open TopologicalSpace
open CategoryTheory
open TopCat
open Opposite
namespace AlgebraicGeometry
structure Scheme extends LocallyRingedSpace where
local_affine :
∀ x : toLocallyRingedSpace,
∃ (U : OpenNhds x) (R : CommRingCat),
Nonempty
(toLocallyRingedSpace.restrict U.openEmbedding ≅ Spec.toLocallyRingedSpace.obj (op R))
#align algebraic_geometry.Scheme AlgebraicGeometry.Scheme
namespace Scheme
-- @[nolint has_nonempty_instance] -- Porting note(#5171): linter not ported yet
def Hom (X Y : Scheme) : Type* :=
X.toLocallyRingedSpace ⟶ Y.toLocallyRingedSpace
#align algebraic_geometry.Scheme.hom AlgebraicGeometry.Scheme.Hom
instance : Category Scheme :=
{ InducedCategory.category Scheme.toLocallyRingedSpace with Hom := Hom }
-- porting note (#10688): added to ease automation
@[continuity]
lemma Hom.continuous {X Y : Scheme} (f : X ⟶ Y) : Continuous f.1.base := f.1.base.2
protected abbrev sheaf (X : Scheme) :=
X.toSheafedSpace.sheaf
#align algebraic_geometry.Scheme.sheaf AlgebraicGeometry.Scheme.sheaf
instance : CoeSort Scheme Type* where
coe X := X.carrier
@[simps!]
def forgetToLocallyRingedSpace : Scheme ⥤ LocallyRingedSpace :=
inducedFunctor _
-- deriving Full, Faithful -- Porting note: no delta derive handler, see https://github.com/leanprover-community/mathlib4/issues/5020
#align algebraic_geometry.Scheme.forget_to_LocallyRingedSpace AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace
@[simps!]
def fullyFaithfulForgetToLocallyRingedSpace :
forgetToLocallyRingedSpace.FullyFaithful :=
fullyFaithfulInducedFunctor _
instance : forgetToLocallyRingedSpace.Full :=
InducedCategory.full _
instance : forgetToLocallyRingedSpace.Faithful :=
InducedCategory.faithful _
@[simps!]
def forgetToTop : Scheme ⥤ TopCat :=
Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToTop
#align algebraic_geometry.Scheme.forget_to_Top AlgebraicGeometry.Scheme.forgetToTop
-- Porting note: Lean seems not able to find this coercion any more
instance hasCoeToTopCat : CoeOut Scheme TopCat where
coe X := X.carrier
-- Porting note: added this unification hint just in case
unif_hint forgetToTop_obj_eq_coe (X : Scheme) where ⊢
forgetToTop.obj X ≟ (X : TopCat)
@[simp]
theorem id_val_base (X : Scheme) : (𝟙 X : _).1.base = 𝟙 _ :=
rfl
#align algebraic_geometry.Scheme.id_val_base AlgebraicGeometry.Scheme.id_val_base
@[simp]
theorem id_app {X : Scheme} (U : (Opens X.carrier)ᵒᵖ) :
(𝟙 X : _).val.c.app U =
X.presheaf.map (eqToHom (by induction' U with U; cases U; rfl)) :=
PresheafedSpace.id_c_app X.toPresheafedSpace U
#align algebraic_geometry.Scheme.id_app AlgebraicGeometry.Scheme.id_app
@[reassoc]
theorem comp_val {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).val = f.val ≫ g.val :=
rfl
#align algebraic_geometry.Scheme.comp_val AlgebraicGeometry.Scheme.comp_val
@[simp, reassoc] -- reassoc lemma does not need `simp`
theorem comp_coeBase {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base :=
rfl
#align algebraic_geometry.Scheme.comp_coe_base AlgebraicGeometry.Scheme.comp_coeBase
-- Porting note: removed elementwise attribute, as generated lemmas were trivial.
@[reassoc]
theorem comp_val_base {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base :=
rfl
#align algebraic_geometry.Scheme.comp_val_base AlgebraicGeometry.Scheme.comp_val_base
| Mathlib/AlgebraicGeometry/Scheme.lean | 144 | 146 | theorem comp_val_base_apply {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g).val.base x = g.val.base (f.val.base x) := by |
simp
| 158 |
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.Scheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u
noncomputable section
open TopologicalSpace
open CategoryTheory
open TopCat
open Opposite
namespace AlgebraicGeometry
structure Scheme extends LocallyRingedSpace where
local_affine :
∀ x : toLocallyRingedSpace,
∃ (U : OpenNhds x) (R : CommRingCat),
Nonempty
(toLocallyRingedSpace.restrict U.openEmbedding ≅ Spec.toLocallyRingedSpace.obj (op R))
#align algebraic_geometry.Scheme AlgebraicGeometry.Scheme
namespace Scheme
-- @[nolint has_nonempty_instance] -- Porting note(#5171): linter not ported yet
def Hom (X Y : Scheme) : Type* :=
X.toLocallyRingedSpace ⟶ Y.toLocallyRingedSpace
#align algebraic_geometry.Scheme.hom AlgebraicGeometry.Scheme.Hom
instance : Category Scheme :=
{ InducedCategory.category Scheme.toLocallyRingedSpace with Hom := Hom }
-- porting note (#10688): added to ease automation
@[continuity]
lemma Hom.continuous {X Y : Scheme} (f : X ⟶ Y) : Continuous f.1.base := f.1.base.2
protected abbrev sheaf (X : Scheme) :=
X.toSheafedSpace.sheaf
#align algebraic_geometry.Scheme.sheaf AlgebraicGeometry.Scheme.sheaf
instance : CoeSort Scheme Type* where
coe X := X.carrier
@[simps!]
def forgetToLocallyRingedSpace : Scheme ⥤ LocallyRingedSpace :=
inducedFunctor _
-- deriving Full, Faithful -- Porting note: no delta derive handler, see https://github.com/leanprover-community/mathlib4/issues/5020
#align algebraic_geometry.Scheme.forget_to_LocallyRingedSpace AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace
@[simps!]
def fullyFaithfulForgetToLocallyRingedSpace :
forgetToLocallyRingedSpace.FullyFaithful :=
fullyFaithfulInducedFunctor _
instance : forgetToLocallyRingedSpace.Full :=
InducedCategory.full _
instance : forgetToLocallyRingedSpace.Faithful :=
InducedCategory.faithful _
@[simps!]
def forgetToTop : Scheme ⥤ TopCat :=
Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToTop
#align algebraic_geometry.Scheme.forget_to_Top AlgebraicGeometry.Scheme.forgetToTop
-- Porting note: Lean seems not able to find this coercion any more
instance hasCoeToTopCat : CoeOut Scheme TopCat where
coe X := X.carrier
-- Porting note: added this unification hint just in case
unif_hint forgetToTop_obj_eq_coe (X : Scheme) where ⊢
forgetToTop.obj X ≟ (X : TopCat)
@[simp]
theorem id_val_base (X : Scheme) : (𝟙 X : _).1.base = 𝟙 _ :=
rfl
#align algebraic_geometry.Scheme.id_val_base AlgebraicGeometry.Scheme.id_val_base
@[simp]
theorem id_app {X : Scheme} (U : (Opens X.carrier)ᵒᵖ) :
(𝟙 X : _).val.c.app U =
X.presheaf.map (eqToHom (by induction' U with U; cases U; rfl)) :=
PresheafedSpace.id_c_app X.toPresheafedSpace U
#align algebraic_geometry.Scheme.id_app AlgebraicGeometry.Scheme.id_app
@[reassoc]
theorem comp_val {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).val = f.val ≫ g.val :=
rfl
#align algebraic_geometry.Scheme.comp_val AlgebraicGeometry.Scheme.comp_val
@[simp, reassoc] -- reassoc lemma does not need `simp`
theorem comp_coeBase {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base :=
rfl
#align algebraic_geometry.Scheme.comp_coe_base AlgebraicGeometry.Scheme.comp_coeBase
-- Porting note: removed elementwise attribute, as generated lemmas were trivial.
@[reassoc]
theorem comp_val_base {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base :=
rfl
#align algebraic_geometry.Scheme.comp_val_base AlgebraicGeometry.Scheme.comp_val_base
theorem comp_val_base_apply {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g).val.base x = g.val.base (f.val.base x) := by
simp
#align algebraic_geometry.Scheme.comp_val_base_apply AlgebraicGeometry.Scheme.comp_val_base_apply
@[simp, reassoc] -- reassoc lemma does not need `simp`
theorem comp_val_c_app {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(f ≫ g).val.c.app U = g.val.c.app U ≫ f.val.c.app _ :=
rfl
#align algebraic_geometry.Scheme.comp_val_c_app AlgebraicGeometry.Scheme.comp_val_c_app
| Mathlib/AlgebraicGeometry/Scheme.lean | 155 | 157 | theorem congr_app {X Y : Scheme} {f g : X ⟶ Y} (e : f = g) (U) :
f.val.c.app U = g.val.c.app U ≫ X.presheaf.map (eqToHom (by subst e; rfl)) := by |
subst e; dsimp; simp
| 158 |
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.Scheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u
noncomputable section
open TopologicalSpace
open CategoryTheory
open TopCat
open Opposite
namespace AlgebraicGeometry
structure Scheme extends LocallyRingedSpace where
local_affine :
∀ x : toLocallyRingedSpace,
∃ (U : OpenNhds x) (R : CommRingCat),
Nonempty
(toLocallyRingedSpace.restrict U.openEmbedding ≅ Spec.toLocallyRingedSpace.obj (op R))
#align algebraic_geometry.Scheme AlgebraicGeometry.Scheme
namespace Scheme
-- @[nolint has_nonempty_instance] -- Porting note(#5171): linter not ported yet
def Hom (X Y : Scheme) : Type* :=
X.toLocallyRingedSpace ⟶ Y.toLocallyRingedSpace
#align algebraic_geometry.Scheme.hom AlgebraicGeometry.Scheme.Hom
instance : Category Scheme :=
{ InducedCategory.category Scheme.toLocallyRingedSpace with Hom := Hom }
-- porting note (#10688): added to ease automation
@[continuity]
lemma Hom.continuous {X Y : Scheme} (f : X ⟶ Y) : Continuous f.1.base := f.1.base.2
protected abbrev sheaf (X : Scheme) :=
X.toSheafedSpace.sheaf
#align algebraic_geometry.Scheme.sheaf AlgebraicGeometry.Scheme.sheaf
instance : CoeSort Scheme Type* where
coe X := X.carrier
@[simps!]
def forgetToLocallyRingedSpace : Scheme ⥤ LocallyRingedSpace :=
inducedFunctor _
-- deriving Full, Faithful -- Porting note: no delta derive handler, see https://github.com/leanprover-community/mathlib4/issues/5020
#align algebraic_geometry.Scheme.forget_to_LocallyRingedSpace AlgebraicGeometry.Scheme.forgetToLocallyRingedSpace
@[simps!]
def fullyFaithfulForgetToLocallyRingedSpace :
forgetToLocallyRingedSpace.FullyFaithful :=
fullyFaithfulInducedFunctor _
instance : forgetToLocallyRingedSpace.Full :=
InducedCategory.full _
instance : forgetToLocallyRingedSpace.Faithful :=
InducedCategory.faithful _
@[simps!]
def forgetToTop : Scheme ⥤ TopCat :=
Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToTop
#align algebraic_geometry.Scheme.forget_to_Top AlgebraicGeometry.Scheme.forgetToTop
-- Porting note: Lean seems not able to find this coercion any more
instance hasCoeToTopCat : CoeOut Scheme TopCat where
coe X := X.carrier
-- Porting note: added this unification hint just in case
unif_hint forgetToTop_obj_eq_coe (X : Scheme) where ⊢
forgetToTop.obj X ≟ (X : TopCat)
@[simp]
theorem id_val_base (X : Scheme) : (𝟙 X : _).1.base = 𝟙 _ :=
rfl
#align algebraic_geometry.Scheme.id_val_base AlgebraicGeometry.Scheme.id_val_base
@[simp]
theorem id_app {X : Scheme} (U : (Opens X.carrier)ᵒᵖ) :
(𝟙 X : _).val.c.app U =
X.presheaf.map (eqToHom (by induction' U with U; cases U; rfl)) :=
PresheafedSpace.id_c_app X.toPresheafedSpace U
#align algebraic_geometry.Scheme.id_app AlgebraicGeometry.Scheme.id_app
@[reassoc]
theorem comp_val {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).val = f.val ≫ g.val :=
rfl
#align algebraic_geometry.Scheme.comp_val AlgebraicGeometry.Scheme.comp_val
@[simp, reassoc] -- reassoc lemma does not need `simp`
theorem comp_coeBase {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base :=
rfl
#align algebraic_geometry.Scheme.comp_coe_base AlgebraicGeometry.Scheme.comp_coeBase
-- Porting note: removed elementwise attribute, as generated lemmas were trivial.
@[reassoc]
theorem comp_val_base {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g).val.base = f.val.base ≫ g.val.base :=
rfl
#align algebraic_geometry.Scheme.comp_val_base AlgebraicGeometry.Scheme.comp_val_base
theorem comp_val_base_apply {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :
(f ≫ g).val.base x = g.val.base (f.val.base x) := by
simp
#align algebraic_geometry.Scheme.comp_val_base_apply AlgebraicGeometry.Scheme.comp_val_base_apply
@[simp, reassoc] -- reassoc lemma does not need `simp`
theorem comp_val_c_app {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
(f ≫ g).val.c.app U = g.val.c.app U ≫ f.val.c.app _ :=
rfl
#align algebraic_geometry.Scheme.comp_val_c_app AlgebraicGeometry.Scheme.comp_val_c_app
theorem congr_app {X Y : Scheme} {f g : X ⟶ Y} (e : f = g) (U) :
f.val.c.app U = g.val.c.app U ≫ X.presheaf.map (eqToHom (by subst e; rfl)) := by
subst e; dsimp; simp
#align algebraic_geometry.Scheme.congr_app AlgebraicGeometry.Scheme.congr_app
| Mathlib/AlgebraicGeometry/Scheme.lean | 160 | 167 | theorem app_eq {X Y : Scheme} (f : X ⟶ Y) {U V : Opens Y.carrier} (e : U = V) :
f.val.c.app (op U) =
Y.presheaf.map (eqToHom e.symm).op ≫
f.val.c.app (op V) ≫
X.presheaf.map (eqToHom (congr_arg (Opens.map f.val.base).obj e)).op := by |
rw [← IsIso.inv_comp_eq, ← Functor.map_inv, f.val.c.naturality, Presheaf.pushforwardObj_map]
cases e
rfl
| 158 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero
assert_not_exists MulAction
variable {ι κ α β γ : Type*}
open Fin Function
namespace Finset
@[to_additive "`∑ x ∈ s, f x` is the sum of `f x` as `x` ranges over the elements
of the finite set `s`."]
protected def prod [CommMonoid β] (s : Finset α) (f : α → β) : β :=
(s.1.map f).prod
#align finset.prod Finset.prod
#align finset.sum Finset.sum
@[to_additive (attr := simp)]
theorem prod_mk [CommMonoid β] (s : Multiset α) (hs : s.Nodup) (f : α → β) :
(⟨s, hs⟩ : Finset α).prod f = (s.map f).prod :=
rfl
#align finset.prod_mk Finset.prod_mk
#align finset.sum_mk Finset.sum_mk
@[to_additive (attr := simp)]
| Mathlib/Algebra/BigOperators/Group/Finset.lean | 67 | 68 | theorem prod_val [CommMonoid α] (s : Finset α) : s.1.prod = s.prod id := by |
rw [Finset.prod, Multiset.map_id]
| 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operators.pi from "leanprover-community/mathlib"@"fa2309577c7009ea243cffdf990cd6c84f0ad497"
@[to_additive (attr := simp)]
theorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)
(s : Finset γ) (g : γ → ∀ a, β a) : (∏ c ∈ s, g c) a = ∏ c ∈ s, g c a :=
map_prod (Pi.evalMonoidHom β a) _ _
#align finset.prod_apply Finset.prod_apply
#align finset.sum_apply Finset.sum_apply
@[to_additive "An 'unapplied' analogue of `Finset.sum_apply`."]
theorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)
(g : γ → ∀ a, β a) : ∏ c ∈ s, g c = fun a ↦ ∏ c ∈ s, g c a :=
funext fun _ ↦ Finset.prod_apply _ _ _
#align finset.prod_fn Finset.prod_fn
#align finset.sum_fn Finset.sum_fn
@[to_additive]
theorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]
[∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=
Finset.prod_apply a Finset.univ g
#align fintype.prod_apply Fintype.prod_apply
#align fintype.sum_apply Fintype.sum_apply
@[to_additive prod_mk_sum]
theorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)
(g : γ → β) : (∏ x ∈ s, f x, ∏ x ∈ s, g x) = ∏ x ∈ s, (f x, g x) :=
haveI := Classical.decEq γ
Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])
#align prod_mk_prod prod_mk_prod
#align prod_mk_sum prod_mk_sum
| Mathlib/Algebra/BigOperators/Pi.lean | 69 | 72 | theorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]
(x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by |
ext
simp
| 160 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operators.pi from "leanprover-community/mathlib"@"fa2309577c7009ea243cffdf990cd6c84f0ad497"
@[to_additive (attr := simp)]
theorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)
(s : Finset γ) (g : γ → ∀ a, β a) : (∏ c ∈ s, g c) a = ∏ c ∈ s, g c a :=
map_prod (Pi.evalMonoidHom β a) _ _
#align finset.prod_apply Finset.prod_apply
#align finset.sum_apply Finset.sum_apply
@[to_additive "An 'unapplied' analogue of `Finset.sum_apply`."]
theorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)
(g : γ → ∀ a, β a) : ∏ c ∈ s, g c = fun a ↦ ∏ c ∈ s, g c a :=
funext fun _ ↦ Finset.prod_apply _ _ _
#align finset.prod_fn Finset.prod_fn
#align finset.sum_fn Finset.sum_fn
@[to_additive]
theorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]
[∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=
Finset.prod_apply a Finset.univ g
#align fintype.prod_apply Fintype.prod_apply
#align fintype.sum_apply Fintype.sum_apply
@[to_additive prod_mk_sum]
theorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)
(g : γ → β) : (∏ x ∈ s, f x, ∏ x ∈ s, g x) = ∏ x ∈ s, (f x, g x) :=
haveI := Classical.decEq γ
Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])
#align prod_mk_prod prod_mk_prod
#align prod_mk_sum prod_mk_sum
theorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]
(x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by
ext
simp
#align pi_eq_sum_univ pi_eq_sum_univ
section MulSingle
variable {I : Type*} [DecidableEq I] {Z : I → Type*}
variable [∀ i, CommMonoid (Z i)]
@[to_additive]
| Mathlib/Algebra/BigOperators/Pi.lean | 81 | 84 | theorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :
(∏ i, Pi.mulSingle i (f i)) = f := by |
ext a
simp
| 160 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operators.pi from "leanprover-community/mathlib"@"fa2309577c7009ea243cffdf990cd6c84f0ad497"
@[to_additive (attr := simp)]
theorem Finset.prod_apply {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (a : α)
(s : Finset γ) (g : γ → ∀ a, β a) : (∏ c ∈ s, g c) a = ∏ c ∈ s, g c a :=
map_prod (Pi.evalMonoidHom β a) _ _
#align finset.prod_apply Finset.prod_apply
#align finset.sum_apply Finset.sum_apply
@[to_additive "An 'unapplied' analogue of `Finset.sum_apply`."]
theorem Finset.prod_fn {α : Type*} {β : α → Type*} {γ} [∀ a, CommMonoid (β a)] (s : Finset γ)
(g : γ → ∀ a, β a) : ∏ c ∈ s, g c = fun a ↦ ∏ c ∈ s, g c a :=
funext fun _ ↦ Finset.prod_apply _ _ _
#align finset.prod_fn Finset.prod_fn
#align finset.sum_fn Finset.sum_fn
@[to_additive]
theorem Fintype.prod_apply {α : Type*} {β : α → Type*} {γ : Type*} [Fintype γ]
[∀ a, CommMonoid (β a)] (a : α) (g : γ → ∀ a, β a) : (∏ c, g c) a = ∏ c, g c a :=
Finset.prod_apply a Finset.univ g
#align fintype.prod_apply Fintype.prod_apply
#align fintype.sum_apply Fintype.sum_apply
@[to_additive prod_mk_sum]
theorem prod_mk_prod {α β γ : Type*} [CommMonoid α] [CommMonoid β] (s : Finset γ) (f : γ → α)
(g : γ → β) : (∏ x ∈ s, f x, ∏ x ∈ s, g x) = ∏ x ∈ s, (f x, g x) :=
haveI := Classical.decEq γ
Finset.induction_on s rfl (by simp (config := { contextual := true }) [Prod.ext_iff])
#align prod_mk_prod prod_mk_prod
#align prod_mk_sum prod_mk_sum
theorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]
(x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by
ext
simp
#align pi_eq_sum_univ pi_eq_sum_univ
section MulSingle
variable {I : Type*} [DecidableEq I] {Z : I → Type*}
variable [∀ i, CommMonoid (Z i)]
@[to_additive]
theorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :
(∏ i, Pi.mulSingle i (f i)) = f := by
ext a
simp
#align finset.univ_prod_mul_single Finset.univ_prod_mulSingle
#align finset.univ_sum_single Finset.univ_sum_single
@[to_additive]
| Mathlib/Algebra/BigOperators/Pi.lean | 89 | 94 | theorem MonoidHom.functions_ext [Finite I] (G : Type*) [CommMonoid G] (g h : (∀ i, Z i) →* G)
(H : ∀ i x, g (Pi.mulSingle i x) = h (Pi.mulSingle i x)) : g = h := by |
cases nonempty_fintype I
ext k
rw [← Finset.univ_prod_mulSingle k, map_prod, map_prod]
simp only [H]
| 160 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Module.Pi
#align_import data.holor from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u
open List
def HolorIndex (ds : List ℕ) : Type :=
{ is : List ℕ // Forall₂ (· < ·) is ds }
#align holor_index HolorIndex
namespace HolorIndex
variable {ds₁ ds₂ ds₃ : List ℕ}
def take : ∀ {ds₁ : List ℕ}, HolorIndex (ds₁ ++ ds₂) → HolorIndex ds₁
| ds, is => ⟨List.take (length ds) is.1, forall₂_take_append is.1 ds ds₂ is.2⟩
#align holor_index.take HolorIndex.take
def drop : ∀ {ds₁ : List ℕ}, HolorIndex (ds₁ ++ ds₂) → HolorIndex ds₂
| ds, is => ⟨List.drop (length ds) is.1, forall₂_drop_append is.1 ds ds₂ is.2⟩
#align holor_index.drop HolorIndex.drop
| Mathlib/Data/Holor.lean | 58 | 59 | theorem cast_type (is : List ℕ) (eq : ds₁ = ds₂) (h : Forall₂ (· < ·) is ds₁) :
(cast (congr_arg HolorIndex eq) ⟨is, h⟩).val = is := by | subst eq; rfl
| 161 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoid A] {t : Set A}
structure IsAddSubmonoid (s : Set A) : Prop where
zero_mem : (0 : A) ∈ s
add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s
#align is_add_submonoid IsAddSubmonoid
@[to_additive]
structure IsSubmonoid (s : Set M) : Prop where
one_mem : (1 : M) ∈ s
mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s
#align is_submonoid IsSubmonoid
theorem Additive.isAddSubmonoid {s : Set M} :
IsSubmonoid s → @IsAddSubmonoid (Additive M) _ s
| ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
#align additive.is_add_submonoid Additive.isAddSubmonoid
theorem Additive.isAddSubmonoid_iff {s : Set M} :
@IsAddSubmonoid (Additive M) _ s ↔ IsSubmonoid s :=
⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Additive.isAddSubmonoid⟩
#align additive.is_add_submonoid_iff Additive.isAddSubmonoid_iff
theorem Multiplicative.isSubmonoid {s : Set A} :
IsAddSubmonoid s → @IsSubmonoid (Multiplicative A) _ s
| ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
#align multiplicative.is_submonoid Multiplicative.isSubmonoid
theorem Multiplicative.isSubmonoid_iff {s : Set A} :
@IsSubmonoid (Multiplicative A) _ s ↔ IsAddSubmonoid s :=
⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Multiplicative.isSubmonoid⟩
#align multiplicative.is_submonoid_iff Multiplicative.isSubmonoid_iff
@[to_additive
"The intersection of two `AddSubmonoid`s of an `AddMonoid` `M` is an `AddSubmonoid` of M."]
theorem IsSubmonoid.inter {s₁ s₂ : Set M} (is₁ : IsSubmonoid s₁) (is₂ : IsSubmonoid s₂) :
IsSubmonoid (s₁ ∩ s₂) :=
{ one_mem := ⟨is₁.one_mem, is₂.one_mem⟩
mul_mem := @fun _ _ hx hy => ⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }
#align is_submonoid.inter IsSubmonoid.inter
#align is_add_submonoid.inter IsAddSubmonoid.inter
@[to_additive
"The intersection of an indexed set of `AddSubmonoid`s of an `AddMonoid` `M` is
an `AddSubmonoid` of `M`."]
theorem IsSubmonoid.iInter {ι : Sort*} {s : ι → Set M} (h : ∀ y : ι, IsSubmonoid (s y)) :
IsSubmonoid (Set.iInter s) :=
{ one_mem := Set.mem_iInter.2 fun y => (h y).one_mem
mul_mem := fun h₁ h₂ =>
Set.mem_iInter.2 fun y => (h y).mul_mem (Set.mem_iInter.1 h₁ y) (Set.mem_iInter.1 h₂ y) }
#align is_submonoid.Inter IsSubmonoid.iInter
#align is_add_submonoid.Inter IsAddSubmonoid.iInter
@[to_additive
"The union of an indexed, directed, nonempty set of `AddSubmonoid`s of an `AddMonoid` `M`
is an `AddSubmonoid` of `M`. "]
theorem isSubmonoid_iUnion_of_directed {ι : Type*} [hι : Nonempty ι] {s : ι → Set M}
(hs : ∀ i, IsSubmonoid (s i)) (Directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubmonoid (⋃ i, s i) :=
{ one_mem :=
let ⟨i⟩ := hι
Set.mem_iUnion.2 ⟨i, (hs i).one_mem⟩
mul_mem := fun ha hb =>
let ⟨i, hi⟩ := Set.mem_iUnion.1 ha
let ⟨j, hj⟩ := Set.mem_iUnion.1 hb
let ⟨k, hk⟩ := Directed i j
Set.mem_iUnion.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }
#align is_submonoid_Union_of_directed isSubmonoid_iUnion_of_directed
#align is_add_submonoid_Union_of_directed isAddSubmonoid_iUnion_of_directed
namespace IsSubmonoid
@[to_additive
"The sum of a list of elements of an `AddSubmonoid` is an element of the `AddSubmonoid`."]
| Mathlib/Deprecated/Submonoid.lean | 232 | 237 | theorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s
| [], _ => hs.one_mem
| a :: l, h =>
suffices a * l.prod ∈ s by simpa
have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by | simpa using h
hs.mul_mem this.1 (list_prod_mem hs this.2)
| 162 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoid A] {t : Set A}
structure IsAddSubmonoid (s : Set A) : Prop where
zero_mem : (0 : A) ∈ s
add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s
#align is_add_submonoid IsAddSubmonoid
@[to_additive]
structure IsSubmonoid (s : Set M) : Prop where
one_mem : (1 : M) ∈ s
mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s
#align is_submonoid IsSubmonoid
theorem Additive.isAddSubmonoid {s : Set M} :
IsSubmonoid s → @IsAddSubmonoid (Additive M) _ s
| ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
#align additive.is_add_submonoid Additive.isAddSubmonoid
theorem Additive.isAddSubmonoid_iff {s : Set M} :
@IsAddSubmonoid (Additive M) _ s ↔ IsSubmonoid s :=
⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Additive.isAddSubmonoid⟩
#align additive.is_add_submonoid_iff Additive.isAddSubmonoid_iff
theorem Multiplicative.isSubmonoid {s : Set A} :
IsAddSubmonoid s → @IsSubmonoid (Multiplicative A) _ s
| ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
#align multiplicative.is_submonoid Multiplicative.isSubmonoid
theorem Multiplicative.isSubmonoid_iff {s : Set A} :
@IsSubmonoid (Multiplicative A) _ s ↔ IsAddSubmonoid s :=
⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Multiplicative.isSubmonoid⟩
#align multiplicative.is_submonoid_iff Multiplicative.isSubmonoid_iff
@[to_additive
"The intersection of two `AddSubmonoid`s of an `AddMonoid` `M` is an `AddSubmonoid` of M."]
theorem IsSubmonoid.inter {s₁ s₂ : Set M} (is₁ : IsSubmonoid s₁) (is₂ : IsSubmonoid s₂) :
IsSubmonoid (s₁ ∩ s₂) :=
{ one_mem := ⟨is₁.one_mem, is₂.one_mem⟩
mul_mem := @fun _ _ hx hy => ⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }
#align is_submonoid.inter IsSubmonoid.inter
#align is_add_submonoid.inter IsAddSubmonoid.inter
@[to_additive
"The intersection of an indexed set of `AddSubmonoid`s of an `AddMonoid` `M` is
an `AddSubmonoid` of `M`."]
theorem IsSubmonoid.iInter {ι : Sort*} {s : ι → Set M} (h : ∀ y : ι, IsSubmonoid (s y)) :
IsSubmonoid (Set.iInter s) :=
{ one_mem := Set.mem_iInter.2 fun y => (h y).one_mem
mul_mem := fun h₁ h₂ =>
Set.mem_iInter.2 fun y => (h y).mul_mem (Set.mem_iInter.1 h₁ y) (Set.mem_iInter.1 h₂ y) }
#align is_submonoid.Inter IsSubmonoid.iInter
#align is_add_submonoid.Inter IsAddSubmonoid.iInter
@[to_additive
"The union of an indexed, directed, nonempty set of `AddSubmonoid`s of an `AddMonoid` `M`
is an `AddSubmonoid` of `M`. "]
theorem isSubmonoid_iUnion_of_directed {ι : Type*} [hι : Nonempty ι] {s : ι → Set M}
(hs : ∀ i, IsSubmonoid (s i)) (Directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubmonoid (⋃ i, s i) :=
{ one_mem :=
let ⟨i⟩ := hι
Set.mem_iUnion.2 ⟨i, (hs i).one_mem⟩
mul_mem := fun ha hb =>
let ⟨i, hi⟩ := Set.mem_iUnion.1 ha
let ⟨j, hj⟩ := Set.mem_iUnion.1 hb
let ⟨k, hk⟩ := Directed i j
Set.mem_iUnion.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }
#align is_submonoid_Union_of_directed isSubmonoid_iUnion_of_directed
#align is_add_submonoid_Union_of_directed isAddSubmonoid_iUnion_of_directed
namespace IsSubmonoid
@[to_additive
"The sum of a list of elements of an `AddSubmonoid` is an element of the `AddSubmonoid`."]
theorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s
| [], _ => hs.one_mem
| a :: l, h =>
suffices a * l.prod ∈ s by simpa
have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by simpa using h
hs.mul_mem this.1 (list_prod_mem hs this.2)
#align is_submonoid.list_prod_mem IsSubmonoid.list_prod_mem
#align is_add_submonoid.list_sum_mem IsAddSubmonoid.list_sum_mem
@[to_additive
"The sum of a multiset of elements of an `AddSubmonoid` of an `AddCommMonoid`
is an element of the `AddSubmonoid`. "]
| Mathlib/Deprecated/Submonoid.lean | 246 | 250 | theorem multiset_prod_mem {M} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (m : Multiset M) :
(∀ a ∈ m, a ∈ s) → m.prod ∈ s := by |
refine Quotient.inductionOn m fun l hl => ?_
rw [Multiset.quot_mk_to_coe, Multiset.prod_coe]
exact list_prod_mem hs hl
| 162 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoid A] {t : Set A}
structure IsAddSubmonoid (s : Set A) : Prop where
zero_mem : (0 : A) ∈ s
add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s
#align is_add_submonoid IsAddSubmonoid
@[to_additive]
structure IsSubmonoid (s : Set M) : Prop where
one_mem : (1 : M) ∈ s
mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s
#align is_submonoid IsSubmonoid
theorem Additive.isAddSubmonoid {s : Set M} :
IsSubmonoid s → @IsAddSubmonoid (Additive M) _ s
| ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
#align additive.is_add_submonoid Additive.isAddSubmonoid
theorem Additive.isAddSubmonoid_iff {s : Set M} :
@IsAddSubmonoid (Additive M) _ s ↔ IsSubmonoid s :=
⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Additive.isAddSubmonoid⟩
#align additive.is_add_submonoid_iff Additive.isAddSubmonoid_iff
theorem Multiplicative.isSubmonoid {s : Set A} :
IsAddSubmonoid s → @IsSubmonoid (Multiplicative A) _ s
| ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
#align multiplicative.is_submonoid Multiplicative.isSubmonoid
theorem Multiplicative.isSubmonoid_iff {s : Set A} :
@IsSubmonoid (Multiplicative A) _ s ↔ IsAddSubmonoid s :=
⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Multiplicative.isSubmonoid⟩
#align multiplicative.is_submonoid_iff Multiplicative.isSubmonoid_iff
@[to_additive
"The intersection of two `AddSubmonoid`s of an `AddMonoid` `M` is an `AddSubmonoid` of M."]
theorem IsSubmonoid.inter {s₁ s₂ : Set M} (is₁ : IsSubmonoid s₁) (is₂ : IsSubmonoid s₂) :
IsSubmonoid (s₁ ∩ s₂) :=
{ one_mem := ⟨is₁.one_mem, is₂.one_mem⟩
mul_mem := @fun _ _ hx hy => ⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }
#align is_submonoid.inter IsSubmonoid.inter
#align is_add_submonoid.inter IsAddSubmonoid.inter
@[to_additive
"The intersection of an indexed set of `AddSubmonoid`s of an `AddMonoid` `M` is
an `AddSubmonoid` of `M`."]
theorem IsSubmonoid.iInter {ι : Sort*} {s : ι → Set M} (h : ∀ y : ι, IsSubmonoid (s y)) :
IsSubmonoid (Set.iInter s) :=
{ one_mem := Set.mem_iInter.2 fun y => (h y).one_mem
mul_mem := fun h₁ h₂ =>
Set.mem_iInter.2 fun y => (h y).mul_mem (Set.mem_iInter.1 h₁ y) (Set.mem_iInter.1 h₂ y) }
#align is_submonoid.Inter IsSubmonoid.iInter
#align is_add_submonoid.Inter IsAddSubmonoid.iInter
@[to_additive
"The union of an indexed, directed, nonempty set of `AddSubmonoid`s of an `AddMonoid` `M`
is an `AddSubmonoid` of `M`. "]
theorem isSubmonoid_iUnion_of_directed {ι : Type*} [hι : Nonempty ι] {s : ι → Set M}
(hs : ∀ i, IsSubmonoid (s i)) (Directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubmonoid (⋃ i, s i) :=
{ one_mem :=
let ⟨i⟩ := hι
Set.mem_iUnion.2 ⟨i, (hs i).one_mem⟩
mul_mem := fun ha hb =>
let ⟨i, hi⟩ := Set.mem_iUnion.1 ha
let ⟨j, hj⟩ := Set.mem_iUnion.1 hb
let ⟨k, hk⟩ := Directed i j
Set.mem_iUnion.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }
#align is_submonoid_Union_of_directed isSubmonoid_iUnion_of_directed
#align is_add_submonoid_Union_of_directed isAddSubmonoid_iUnion_of_directed
@[to_additive "Create a bundled additive submonoid from a set `s` and `[IsAddSubmonoid s]`."]
def Submonoid.of {s : Set M} (h : IsSubmonoid s) : Submonoid M :=
⟨⟨s, @fun _ _ => h.2⟩, h.1⟩
#align submonoid.of Submonoid.of
#align add_submonoid.of AddSubmonoid.of
@[to_additive]
| Mathlib/Deprecated/Submonoid.lean | 426 | 427 | theorem Submonoid.isSubmonoid (S : Submonoid M) : IsSubmonoid (S : Set M) := by |
exact ⟨S.2, S.1.2⟩
| 162 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
| Mathlib/Data/Nat/GCD/BigOperators.lean | 20 | 22 | theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by |
induction l <;> simp [Nat.coprime_mul_iff_left, *]
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
| Mathlib/Data/Nat/GCD/BigOperators.lean | 24 | 26 | theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by |
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
| Mathlib/Data/Nat/GCD/BigOperators.lean | 28 | 30 | theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
| Mathlib/Data/Nat/GCD/BigOperators.lean | 32 | 34 | theorem coprime_multiset_prod_right_iff {k : ℕ} {m : Multiset ℕ} :
Coprime k m.prod ↔ ∀ n ∈ m, Coprime k n := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
theorem coprime_multiset_prod_right_iff {k : ℕ} {m : Multiset ℕ} :
Coprime k m.prod ↔ ∀ n ∈ m, Coprime k n := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
| Mathlib/Data/Nat/GCD/BigOperators.lean | 36 | 38 | theorem coprime_prod_left_iff {t : Finset ι} {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, Coprime (s i) x := by |
simpa using coprime_multiset_prod_left_iff (m := t.val.map s)
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
theorem coprime_multiset_prod_right_iff {k : ℕ} {m : Multiset ℕ} :
Coprime k m.prod ↔ ∀ n ∈ m, Coprime k n := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
theorem coprime_prod_left_iff {t : Finset ι} {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, Coprime (s i) x := by
simpa using coprime_multiset_prod_left_iff (m := t.val.map s)
| Mathlib/Data/Nat/GCD/BigOperators.lean | 40 | 42 | theorem coprime_prod_right_iff {x : ℕ} {t : Finset ι} {s : ι → ℕ} :
Coprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, Coprime x (s i) := by |
simpa using coprime_multiset_prod_right_iff (m := t.val.map s)
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
theorem coprime_multiset_prod_right_iff {k : ℕ} {m : Multiset ℕ} :
Coprime k m.prod ↔ ∀ n ∈ m, Coprime k n := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
theorem coprime_prod_left_iff {t : Finset ι} {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, Coprime (s i) x := by
simpa using coprime_multiset_prod_left_iff (m := t.val.map s)
theorem coprime_prod_right_iff {x : ℕ} {t : Finset ι} {s : ι → ℕ} :
Coprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, Coprime x (s i) := by
simpa using coprime_multiset_prod_right_iff (m := t.val.map s)
alias ⟨_, Coprime.prod_left⟩ := coprime_prod_left_iff
#align nat.coprime_prod_left Nat.Coprime.prod_left
alias ⟨_, Coprime.prod_right⟩ := coprime_prod_right_iff
#align nat.coprime_prod_right Nat.Coprime.prod_right
| Mathlib/Data/Nat/GCD/BigOperators.lean | 52 | 54 | theorem coprime_fintype_prod_left_iff [Fintype ι] {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i, s i) x ↔ ∀ i, Coprime (s i) x := by |
simp [coprime_prod_left_iff]
| 163 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
induction l <;> simp [Nat.coprime_mul_iff_left, *]
theorem coprime_list_prod_right_iff {k : ℕ} {l : List ℕ} :
Coprime k l.prod ↔ ∀ n ∈ l, Coprime k n := by
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
theorem coprime_multiset_prod_left_iff {m : Multiset ℕ} {k : ℕ} :
Coprime m.prod k ↔ ∀ n ∈ m, Coprime n k := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
theorem coprime_multiset_prod_right_iff {k : ℕ} {m : Multiset ℕ} :
Coprime k m.prod ↔ ∀ n ∈ m, Coprime k n := by
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
theorem coprime_prod_left_iff {t : Finset ι} {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, Coprime (s i) x := by
simpa using coprime_multiset_prod_left_iff (m := t.val.map s)
theorem coprime_prod_right_iff {x : ℕ} {t : Finset ι} {s : ι → ℕ} :
Coprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, Coprime x (s i) := by
simpa using coprime_multiset_prod_right_iff (m := t.val.map s)
alias ⟨_, Coprime.prod_left⟩ := coprime_prod_left_iff
#align nat.coprime_prod_left Nat.Coprime.prod_left
alias ⟨_, Coprime.prod_right⟩ := coprime_prod_right_iff
#align nat.coprime_prod_right Nat.Coprime.prod_right
theorem coprime_fintype_prod_left_iff [Fintype ι] {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i, s i) x ↔ ∀ i, Coprime (s i) x := by
simp [coprime_prod_left_iff]
| Mathlib/Data/Nat/GCD/BigOperators.lean | 56 | 58 | theorem coprime_fintype_prod_right_iff [Fintype ι] {x : ℕ} {s : ι → ℕ} :
Coprime x (∏ i, s i) ↔ ∀ i, Coprime x (s i) := by |
simp [coprime_prod_right_iff]
| 163 |
import Mathlib.Data.List.Lex
import Mathlib.Data.Char
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algebra.Order.Group.Nat
#align_import data.string.basic from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
namespace String
def ltb (s₁ s₂ : Iterator) : Bool :=
if s₂.hasNext then
if s₁.hasNext then
if s₁.curr = s₂.curr then
ltb s₁.next s₂.next
else s₁.curr < s₂.curr
else true
else false
#align string.ltb String.ltb
instance LT' : LT String :=
⟨fun s₁ s₂ ↦ ltb s₁.iter s₂.iter⟩
#align string.has_lt' String.LT'
instance decidableLT : @DecidableRel String (· < ·) := by
simp only [LT']
infer_instance -- short-circuit type class inference
#align string.decidable_lt String.decidableLT
def ltb.inductionOn.{u} {motive : Iterator → Iterator → Sort u} (it₁ it₂ : Iterator)
(ind : ∀ s₁ s₂ i₁ i₂, Iterator.hasNext ⟨s₂, i₂⟩ → Iterator.hasNext ⟨s₁, i₁⟩ →
get s₁ i₁ = get s₂ i₂ → motive (Iterator.next ⟨s₁, i₁⟩) (Iterator.next ⟨s₂, i₂⟩) →
motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)
(eq : ∀ s₁ s₂ i₁ i₂, Iterator.hasNext ⟨s₂, i₂⟩ → Iterator.hasNext ⟨s₁, i₁⟩ →
¬ get s₁ i₁ = get s₂ i₂ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)
(base₁ : ∀ s₁ s₂ i₁ i₂, Iterator.hasNext ⟨s₂, i₂⟩ → ¬ Iterator.hasNext ⟨s₁, i₁⟩ →
motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩)
(base₂ : ∀ s₁ s₂ i₁ i₂, ¬ Iterator.hasNext ⟨s₂, i₂⟩ → motive ⟨s₁, i₁⟩ ⟨s₂, i₂⟩) :
motive it₁ it₂ :=
if h₂ : it₂.hasNext then
if h₁ : it₁.hasNext then
if heq : it₁.curr = it₂.curr then
ind it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq (inductionOn it₁.next it₂.next ind eq base₁ base₂)
else eq it₁.s it₂.s it₁.i it₂.i h₂ h₁ heq
else base₁ it₁.s it₂.s it₁.i it₂.i h₂ h₁
else base₂ it₁.s it₂.s it₁.i it₂.i h₂
| Mathlib/Data/String/Basic.lean | 60 | 74 | theorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos) :
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ = ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ := by |
apply ltb.inductionOn ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ (motive := fun ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ ↦
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ =
ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩) <;> simp only <;>
intro ⟨cs₁⟩ ⟨cs₂⟩ i₁ i₂ <;>
intros <;>
(conv => lhs; unfold ltb) <;> (conv => rhs; unfold ltb) <;>
simp only [Iterator.hasNext_cons_addChar, ite_false, ite_true, *]
· rename_i h₂ h₁ heq ih
simp only [Iterator.next, next, heq, Iterator.curr, get_cons_addChar, ite_true] at ih ⊢
repeat rw [Pos.addChar_right_comm _ c]
exact ih
· rename_i h₂ h₁ hne
simp [Iterator.curr, get_cons_addChar, hne]
| 164 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
| Mathlib/Order/Interval/Finset/Nat.lean | 61 | 63 | theorem Iio_eq_range : Iio = range := by |
ext b x
rw [mem_Iio, mem_range]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 67 | 67 | theorem Ico_zero_eq_range : Ico 0 = range := by | rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
#align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by
ext b
simp_all only [mem_Icc, zero_le, true_and, mem_range]
exact lt_iff_le_pred (zero_lt_of_ne_zero hn)
theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
Ico_zero_eq_range.symm
#align finset.range_eq_Ico Finset.range_eq_Ico
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a :=
List.length_range' _ _ _
#align nat.card_Icc Nat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ico Nat.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ioc Nat.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 :=
List.length_range' _ _ _
#align nat.card_Ioo Nat.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 :=
(card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega
#align nat.card_uIcc Nat.card_uIcc
@[simp]
lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iic Nat.card_Iic
@[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 109 | 109 | theorem card_Iio : (Iio b).card = b := by | rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
#align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by
ext b
simp_all only [mem_Icc, zero_le, true_and, mem_range]
exact lt_iff_le_pred (zero_lt_of_ne_zero hn)
theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
Ico_zero_eq_range.symm
#align finset.range_eq_Ico Finset.range_eq_Ico
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a :=
List.length_range' _ _ _
#align nat.card_Icc Nat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ico Nat.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ioc Nat.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 :=
List.length_range' _ _ _
#align nat.card_Ioo Nat.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 :=
(card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega
#align nat.card_uIcc Nat.card_uIcc
@[simp]
lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iic Nat.card_Iic
@[simp]
theorem card_Iio : (Iio b).card = b := by rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iio Nat.card_Iio
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 114 | 115 | theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by |
rw [Fintype.card_ofFinset, card_Icc]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
#align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by
ext b
simp_all only [mem_Icc, zero_le, true_and, mem_range]
exact lt_iff_le_pred (zero_lt_of_ne_zero hn)
theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
Ico_zero_eq_range.symm
#align finset.range_eq_Ico Finset.range_eq_Ico
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a :=
List.length_range' _ _ _
#align nat.card_Icc Nat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ico Nat.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ioc Nat.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 :=
List.length_range' _ _ _
#align nat.card_Ioo Nat.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 :=
(card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega
#align nat.card_uIcc Nat.card_uIcc
@[simp]
lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iic Nat.card_Iic
@[simp]
theorem card_Iio : (Iio b).card = b := by rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iio Nat.card_Iio
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [Fintype.card_ofFinset, card_Icc]
#align nat.card_fintype_Icc Nat.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 120 | 121 | theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ico]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
#align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by
ext b
simp_all only [mem_Icc, zero_le, true_and, mem_range]
exact lt_iff_le_pred (zero_lt_of_ne_zero hn)
theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
Ico_zero_eq_range.symm
#align finset.range_eq_Ico Finset.range_eq_Ico
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a :=
List.length_range' _ _ _
#align nat.card_Icc Nat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ico Nat.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ioc Nat.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 :=
List.length_range' _ _ _
#align nat.card_Ioo Nat.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 :=
(card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega
#align nat.card_uIcc Nat.card_uIcc
@[simp]
lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iic Nat.card_Iic
@[simp]
theorem card_Iio : (Iio b).card = b := by rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iio Nat.card_Iio
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [Fintype.card_ofFinset, card_Icc]
#align nat.card_fintype_Icc Nat.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [Fintype.card_ofFinset, card_Ico]
#align nat.card_fintype_Ico Nat.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 126 | 127 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ioc]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
#align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by
ext b
simp_all only [mem_Icc, zero_le, true_and, mem_range]
exact lt_iff_le_pred (zero_lt_of_ne_zero hn)
theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
Ico_zero_eq_range.symm
#align finset.range_eq_Ico Finset.range_eq_Ico
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a :=
List.length_range' _ _ _
#align nat.card_Icc Nat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ico Nat.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ioc Nat.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 :=
List.length_range' _ _ _
#align nat.card_Ioo Nat.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 :=
(card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega
#align nat.card_uIcc Nat.card_uIcc
@[simp]
lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iic Nat.card_Iic
@[simp]
theorem card_Iio : (Iio b).card = b := by rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iio Nat.card_Iio
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [Fintype.card_ofFinset, card_Icc]
#align nat.card_fintype_Icc Nat.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [Fintype.card_ofFinset, card_Ico]
#align nat.card_fintype_Ico Nat.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by
rw [Fintype.card_ofFinset, card_Ioc]
#align nat.card_fintype_Ioc Nat.card_fintypeIoc
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 132 | 133 | theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by |
rw [Fintype.card_ofFinset, card_Ioo]
| 165 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b := ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩
finsetIco a b := ⟨List.range' a (b - a), List.nodup_range' _ _⟩
finsetIoc a b := ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩
finsetIoo a b := ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩
finset_mem_Icc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ico a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioc a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
finset_mem_Ioo a b x := by rw [Finset.mem_mk, Multiset.mem_coe, List.mem_range'_1]; omega
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Icc_eq_range' Nat.Icc_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ico_eq_range' Nat.Ico_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioc_eq_range' Nat.Ioc_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ :=
rfl
#align nat.Ioo_eq_range' Nat.Ioo_eq_range'
theorem uIcc_eq_range' :
uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ := rfl
#align nat.uIcc_eq_range' Nat.uIcc_eq_range'
theorem Iio_eq_range : Iio = range := by
ext b x
rw [mem_Iio, mem_range]
#align nat.Iio_eq_range Nat.Iio_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range := by rw [← Nat.bot_eq_zero, ← Iio_eq_Ico, Iio_eq_range]
#align nat.Ico_zero_eq_range Nat.Ico_zero_eq_range
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0): range n = Icc 0 (n - 1) := by
ext b
simp_all only [mem_Icc, zero_le, true_and, mem_range]
exact lt_iff_le_pred (zero_lt_of_ne_zero hn)
theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
Ico_zero_eq_range.symm
#align finset.range_eq_Ico Finset.range_eq_Ico
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a :=
List.length_range' _ _ _
#align nat.card_Icc Nat.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ico Nat.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a :=
List.length_range' _ _ _
#align nat.card_Ioc Nat.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 :=
List.length_range' _ _ _
#align nat.card_Ioo Nat.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 :=
(card_Icc _ _).trans $ by rw [← Int.natCast_inj, sup_eq_max, inf_eq_min, Int.ofNat_sub] <;> omega
#align nat.card_uIcc Nat.card_uIcc
@[simp]
lemma card_Iic : (Iic b).card = b + 1 := by rw [Iic_eq_Icc, card_Icc, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iic Nat.card_Iic
@[simp]
theorem card_Iio : (Iio b).card = b := by rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
#align nat.card_Iio Nat.card_Iio
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [Fintype.card_ofFinset, card_Icc]
#align nat.card_fintype_Icc Nat.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [Fintype.card_ofFinset, card_Ico]
#align nat.card_fintype_Ico Nat.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by
rw [Fintype.card_ofFinset, card_Ioc]
#align nat.card_fintype_Ioc Nat.card_fintypeIoc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by
rw [Fintype.card_ofFinset, card_Ioo]
#align nat.card_fintype_Ioo Nat.card_fintypeIoo
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Nat.lean | 138 | 139 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by |
rw [Fintype.card_ofFinset, card_Iic]
| 165 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 79 | 80 | theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by |
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 84 | 85 | theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by |
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 89 | 90 | theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by |
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 94 | 95 | theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by |
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 104 | 105 | theorem card_Icc : (Icc a b).card = b + 1 - a := by |
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 109 | 110 | theorem card_Ico : (Ico a b).card = b - a := by |
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 114 | 115 | theorem card_Ioc : (Ioc a b).card = b - a := by |
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 119 | 120 | theorem card_Ioo : (Ioo a b).card = b - a - 1 := by |
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 124 | 125 | theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by |
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 130 | 131 | theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by |
rw [← card_Icc, Fintype.card_ofFinset]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [← card_Icc, Fintype.card_ofFinset]
#align fin.card_fintype_Icc Fin.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 136 | 137 | theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by |
rw [← card_Ico, Fintype.card_ofFinset]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [← card_Icc, Fintype.card_ofFinset]
#align fin.card_fintype_Icc Fin.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [← card_Ico, Fintype.card_ofFinset]
#align fin.card_fintype_Ico Fin.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 142 | 143 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [← card_Ioc, Fintype.card_ofFinset]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [← card_Icc, Fintype.card_ofFinset]
#align fin.card_fintype_Icc Fin.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [← card_Ico, Fintype.card_ofFinset]
#align fin.card_fintype_Ico Fin.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by
rw [← card_Ioc, Fintype.card_ofFinset]
#align fin.card_fintype_Ioc Fin.card_fintypeIoc
-- Porting note (#10618): simp can prove this
-- @[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 148 | 149 | theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by |
rw [← card_Ioo, Fintype.card_ofFinset]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [← card_Icc, Fintype.card_ofFinset]
#align fin.card_fintype_Icc Fin.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [← card_Ico, Fintype.card_ofFinset]
#align fin.card_fintype_Ico Fin.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by
rw [← card_Ioc, Fintype.card_ofFinset]
#align fin.card_fintype_Ioc Fin.card_fintypeIoc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by
rw [← card_Ioo, Fintype.card_ofFinset]
#align fin.card_fintype_Ioo Fin.card_fintypeIoo
| Mathlib/Order/Interval/Finset/Fin.lean | 152 | 153 | theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a : ℤ).natAbs + 1 := by |
rw [← card_uIcc, Fintype.card_ofFinset]
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [← card_Icc, Fintype.card_ofFinset]
#align fin.card_fintype_Icc Fin.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [← card_Ico, Fintype.card_ofFinset]
#align fin.card_fintype_Ico Fin.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by
rw [← card_Ioc, Fintype.card_ofFinset]
#align fin.card_fintype_Ioc Fin.card_fintypeIoc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by
rw [← card_Ioo, Fintype.card_ofFinset]
#align fin.card_fintype_Ioo Fin.card_fintypeIoo
theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a : ℤ).natAbs + 1 := by
rw [← card_uIcc, Fintype.card_ofFinset]
#align fin.card_fintype_uIcc Fin.card_fintype_uIcc
| Mathlib/Order/Interval/Finset/Fin.lean | 156 | 158 | theorem Ici_eq_finset_subtype : Ici a = (Icc (a : ℕ) n).fin n := by |
ext
simp
| 166 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
OrderIso.locallyFiniteOrder Fin.orderIsoSubtype
instance instLocallyFiniteOrderBot : LocallyFiniteOrderBot (Fin n) :=
OrderIso.locallyFiniteOrderBot Fin.orderIsoSubtype
instance instLocallyFiniteOrderTop : ∀ n, LocallyFiniteOrderTop (Fin n)
| 0 => IsEmpty.toLocallyFiniteOrderTop
| _ + 1 => inferInstance
variable {n} (a b : Fin n)
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n :=
rfl
#align fin.Icc_eq_finset_subtype Fin.Icc_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n :=
rfl
#align fin.Ico_eq_finset_subtype Fin.Ico_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n :=
rfl
#align fin.Ioc_eq_finset_subtype Fin.Ioc_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n :=
rfl
#align fin.Ioo_eq_finset_subtype Fin.Ioo_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n := rfl
#align fin.uIcc_eq_finset_subtype Fin.uIcc_eq_finset_subtype
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Icc Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ico Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc ↑a ↑b := by
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
#align fin.map_subtype_embedding_Ioc Fin.map_valEmbedding_Ioc
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a b).map Fin.valEmbedding = Ioo ↑a ↑b := by
simp [Ioo_eq_finset_subtype, Finset.fin, Finset.map_map]
#align fin.map_subtype_embedding_Ioo Fin.map_valEmbedding_Ioo
@[simp]
theorem map_subtype_embedding_uIcc : (uIcc a b).map valEmbedding = uIcc ↑a ↑b :=
map_valEmbedding_Icc _ _
#align fin.map_subtype_embedding_uIcc Fin.map_subtype_embedding_uIcc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc, ← map_valEmbedding_Icc, card_map]
#align fin.card_Icc Fin.card_Icc
@[simp]
theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico, ← map_valEmbedding_Ico, card_map]
#align fin.card_Ico Fin.card_Ico
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a := by
rw [← Nat.card_Ioc, ← map_valEmbedding_Ioc, card_map]
#align fin.card_Ioc Fin.card_Ioc
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
#align fin.card_Ioo Fin.card_Ioo
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
#align fin.card_uIcc Fin.card_uIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a := by
rw [← card_Icc, Fintype.card_ofFinset]
#align fin.card_fintype_Icc Fin.card_fintypeIcc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by
rw [← card_Ico, Fintype.card_ofFinset]
#align fin.card_fintype_Ico Fin.card_fintypeIco
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by
rw [← card_Ioc, Fintype.card_ofFinset]
#align fin.card_fintype_Ioc Fin.card_fintypeIoc
-- Porting note (#10618): simp can prove this
-- @[simp]
theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by
rw [← card_Ioo, Fintype.card_ofFinset]
#align fin.card_fintype_Ioo Fin.card_fintypeIoo
theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a : ℤ).natAbs + 1 := by
rw [← card_uIcc, Fintype.card_ofFinset]
#align fin.card_fintype_uIcc Fin.card_fintype_uIcc
theorem Ici_eq_finset_subtype : Ici a = (Icc (a : ℕ) n).fin n := by
ext
simp
#align fin.Ici_eq_finset_subtype Fin.Ici_eq_finset_subtype
| Mathlib/Order/Interval/Finset/Fin.lean | 161 | 163 | theorem Ioi_eq_finset_subtype : Ioi a = (Ioc (a : ℕ) n).fin n := by |
ext
simp
| 166 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 61 | 64 | theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by |
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 68 | 72 | theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by |
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
| Mathlib/NumberTheory/Divisors.lean | 75 | 75 | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by | simp [properDivisors]
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 79 | 81 | theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by |
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
| Mathlib/NumberTheory/Divisors.lean | 84 | 86 | theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by |
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
#align nat.insert_self_proper_divisors Nat.insert_self_properDivisors
| Mathlib/NumberTheory/Divisors.lean | 89 | 91 | theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by |
rw [cons_eq_insert, insert_self_properDivisors h]
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
#align nat.insert_self_proper_divisors Nat.insert_self_properDivisors
theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by
rw [cons_eq_insert, insert_self_properDivisors h]
#align nat.cons_self_proper_divisors Nat.cons_self_properDivisors
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 95 | 99 | theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by |
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
#align nat.insert_self_proper_divisors Nat.insert_self_properDivisors
theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by
rw [cons_eq_insert, insert_self_properDivisors h]
#align nat.cons_self_proper_divisors Nat.cons_self_properDivisors
@[simp]
theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
#align nat.mem_divisors Nat.mem_divisors
| Mathlib/NumberTheory/Divisors.lean | 102 | 102 | theorem one_mem_divisors : 1 ∈ divisors n ↔ n ≠ 0 := by | simp
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
#align nat.insert_self_proper_divisors Nat.insert_self_properDivisors
theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by
rw [cons_eq_insert, insert_self_properDivisors h]
#align nat.cons_self_proper_divisors Nat.cons_self_properDivisors
@[simp]
theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
#align nat.mem_divisors Nat.mem_divisors
theorem one_mem_divisors : 1 ∈ divisors n ↔ n ≠ 0 := by simp
#align nat.one_mem_divisors Nat.one_mem_divisors
theorem mem_divisors_self (n : ℕ) (h : n ≠ 0) : n ∈ n.divisors :=
mem_divisors.2 ⟨dvd_rfl, h⟩
#align nat.mem_divisors_self Nat.mem_divisors_self
| Mathlib/NumberTheory/Divisors.lean | 109 | 112 | theorem dvd_of_mem_divisors {m : ℕ} (h : n ∈ divisors m) : n ∣ m := by |
cases m
· apply dvd_zero
· simp [mem_divisors.1 h]
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
#align nat.insert_self_proper_divisors Nat.insert_self_properDivisors
theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by
rw [cons_eq_insert, insert_self_properDivisors h]
#align nat.cons_self_proper_divisors Nat.cons_self_properDivisors
@[simp]
theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
#align nat.mem_divisors Nat.mem_divisors
theorem one_mem_divisors : 1 ∈ divisors n ↔ n ≠ 0 := by simp
#align nat.one_mem_divisors Nat.one_mem_divisors
theorem mem_divisors_self (n : ℕ) (h : n ≠ 0) : n ∈ n.divisors :=
mem_divisors.2 ⟨dvd_rfl, h⟩
#align nat.mem_divisors_self Nat.mem_divisors_self
theorem dvd_of_mem_divisors {m : ℕ} (h : n ∈ divisors m) : n ∣ m := by
cases m
· apply dvd_zero
· simp [mem_divisors.1 h]
#align nat.dvd_of_mem_divisors Nat.dvd_of_mem_divisors
@[simp]
| Mathlib/NumberTheory/Divisors.lean | 116 | 131 | theorem mem_divisorsAntidiagonal {x : ℕ × ℕ} :
x ∈ divisorsAntidiagonal n ↔ x.fst * x.snd = n ∧ n ≠ 0 := by |
simp only [divisorsAntidiagonal, Finset.mem_Ico, Ne, Finset.mem_filter, Finset.mem_product]
rw [and_comm]
apply and_congr_right
rintro rfl
constructor <;> intro h
· contrapose! h
simp [h]
· rw [Nat.lt_add_one_iff, Nat.lt_add_one_iff]
rw [mul_eq_zero, not_or] at h
simp only [succ_le_of_lt (Nat.pos_of_ne_zero h.1), succ_le_of_lt (Nat.pos_of_ne_zero h.2),
true_and_iff]
exact
⟨Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero h.2),
Nat.le_mul_of_pos_left _ (Nat.pos_of_ne_zero h.1)⟩
| 167 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
def divisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 (n + 1))
#align nat.divisors Nat.divisors
def properDivisors : Finset ℕ :=
Finset.filter (fun x : ℕ => x ∣ n) (Finset.Ico 1 n)
#align nat.proper_divisors Nat.properDivisors
def divisorsAntidiagonal : Finset (ℕ × ℕ) :=
Finset.filter (fun x => x.fst * x.snd = n) (Ico 1 (n + 1) ×ˢ Ico 1 (n + 1))
#align nat.divisors_antidiagonal Nat.divisorsAntidiagonal
variable {n}
@[simp]
theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_divisors Nat.filter_dvd_eq_divisors
@[simp]
theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
#align nat.filter_dvd_eq_proper_divisors Nat.filter_dvd_eq_properDivisors
theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by simp [properDivisors]
#align nat.proper_divisors.not_self_mem Nat.properDivisors.not_self_mem
@[simp]
theorem mem_properDivisors {m : ℕ} : n ∈ properDivisors m ↔ n ∣ m ∧ n < m := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [properDivisors]
simp only [and_comm, ← filter_dvd_eq_properDivisors hm, mem_filter, mem_range]
#align nat.mem_proper_divisors Nat.mem_properDivisors
theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
#align nat.insert_self_proper_divisors Nat.insert_self_properDivisors
theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by
rw [cons_eq_insert, insert_self_properDivisors h]
#align nat.cons_self_proper_divisors Nat.cons_self_properDivisors
@[simp]
theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
#align nat.mem_divisors Nat.mem_divisors
theorem one_mem_divisors : 1 ∈ divisors n ↔ n ≠ 0 := by simp
#align nat.one_mem_divisors Nat.one_mem_divisors
theorem mem_divisors_self (n : ℕ) (h : n ≠ 0) : n ∈ n.divisors :=
mem_divisors.2 ⟨dvd_rfl, h⟩
#align nat.mem_divisors_self Nat.mem_divisors_self
theorem dvd_of_mem_divisors {m : ℕ} (h : n ∈ divisors m) : n ∣ m := by
cases m
· apply dvd_zero
· simp [mem_divisors.1 h]
#align nat.dvd_of_mem_divisors Nat.dvd_of_mem_divisors
@[simp]
theorem mem_divisorsAntidiagonal {x : ℕ × ℕ} :
x ∈ divisorsAntidiagonal n ↔ x.fst * x.snd = n ∧ n ≠ 0 := by
simp only [divisorsAntidiagonal, Finset.mem_Ico, Ne, Finset.mem_filter, Finset.mem_product]
rw [and_comm]
apply and_congr_right
rintro rfl
constructor <;> intro h
· contrapose! h
simp [h]
· rw [Nat.lt_add_one_iff, Nat.lt_add_one_iff]
rw [mul_eq_zero, not_or] at h
simp only [succ_le_of_lt (Nat.pos_of_ne_zero h.1), succ_le_of_lt (Nat.pos_of_ne_zero h.2),
true_and_iff]
exact
⟨Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero h.2),
Nat.le_mul_of_pos_left _ (Nat.pos_of_ne_zero h.1)⟩
#align nat.mem_divisors_antidiagonal Nat.mem_divisorsAntidiagonal
lemma ne_zero_of_mem_divisorsAntidiagonal {p : ℕ × ℕ} (hp : p ∈ n.divisorsAntidiagonal) :
p.1 ≠ 0 ∧ p.2 ≠ 0 := by
obtain ⟨hp₁, hp₂⟩ := Nat.mem_divisorsAntidiagonal.mp hp
exact mul_ne_zero_iff.mp (hp₁.symm ▸ hp₂)
lemma left_ne_zero_of_mem_divisorsAntidiagonal {p : ℕ × ℕ} (hp : p ∈ n.divisorsAntidiagonal) :
p.1 ≠ 0 :=
(ne_zero_of_mem_divisorsAntidiagonal hp).1
lemma right_ne_zero_of_mem_divisorsAntidiagonal {p : ℕ × ℕ} (hp : p ∈ n.divisorsAntidiagonal) :
p.2 ≠ 0 :=
(ne_zero_of_mem_divisorsAntidiagonal hp).2
| Mathlib/NumberTheory/Divisors.lean | 147 | 151 | theorem divisor_le {m : ℕ} : n ∈ divisors m → n ≤ m := by |
cases' m with m
· simp
· simp only [mem_divisors, Nat.succ_ne_zero m, and_true_iff, Ne, not_false_iff]
exact Nat.le_of_dvd (Nat.succ_pos m)
| 167 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
section Sigma
variable {α α₁ α₂ : Type*} {β : α → Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
namespace Sigma
instance instInhabitedSigma [Inhabited α] [Inhabited (β default)] : Inhabited (Sigma β) :=
⟨⟨default, default⟩⟩
instance instDecidableEqSigma [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] :
DecidableEq (Sigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFalse n => isFalse fun h ↦ Sigma.noConfusion h fun _ e₂ ↦ n <| eq_of_heq e₂
| _, _, _, _, isFalse n => isFalse fun h ↦ Sigma.noConfusion h fun e₁ _ ↦ n e₁
-- sometimes the built-in injectivity support does not work
@[simp] -- @[nolint simpNF]
theorem mk.inj_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :
Sigma.mk a₁ b₁ = ⟨a₂, b₂⟩ ↔ a₁ = a₂ ∧ HEq b₁ b₂ :=
⟨fun h ↦ by cases h; simp,
fun ⟨h₁, h₂⟩ ↦ by subst h₁; rw [eq_of_heq h₂]⟩
#align sigma.mk.inj_iff Sigma.mk.inj_iff
@[simp]
theorem eta : ∀ x : Σa, β a, Sigma.mk x.1 x.2 = x
| ⟨_, _⟩ => rfl
#align sigma.eta Sigma.eta
#align sigma.ext Sigma.ext
| Mathlib/Data/Sigma/Basic.lean | 70 | 71 | theorem ext_iff {x₀ x₁ : Sigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by |
cases x₀; cases x₁; exact Sigma.mk.inj_iff
| 168 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
section Sigma
variable {α α₁ α₂ : Type*} {β : α → Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
namespace Sigma
instance instInhabitedSigma [Inhabited α] [Inhabited (β default)] : Inhabited (Sigma β) :=
⟨⟨default, default⟩⟩
instance instDecidableEqSigma [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] :
DecidableEq (Sigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFalse n => isFalse fun h ↦ Sigma.noConfusion h fun _ e₂ ↦ n <| eq_of_heq e₂
| _, _, _, _, isFalse n => isFalse fun h ↦ Sigma.noConfusion h fun e₁ _ ↦ n e₁
-- sometimes the built-in injectivity support does not work
@[simp] -- @[nolint simpNF]
theorem mk.inj_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :
Sigma.mk a₁ b₁ = ⟨a₂, b₂⟩ ↔ a₁ = a₂ ∧ HEq b₁ b₂ :=
⟨fun h ↦ by cases h; simp,
fun ⟨h₁, h₂⟩ ↦ by subst h₁; rw [eq_of_heq h₂]⟩
#align sigma.mk.inj_iff Sigma.mk.inj_iff
@[simp]
theorem eta : ∀ x : Σa, β a, Sigma.mk x.1 x.2 = x
| ⟨_, _⟩ => rfl
#align sigma.eta Sigma.eta
#align sigma.ext Sigma.ext
theorem ext_iff {x₀ x₁ : Sigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by
cases x₀; cases x₁; exact Sigma.mk.inj_iff
#align sigma.ext_iff Sigma.ext_iff
| Mathlib/Data/Sigma/Basic.lean | 75 | 80 | theorem _root_.Function.eq_of_sigmaMk_comp {γ : Type*} [Nonempty γ]
{a b : α} {f : γ → β a} {g : γ → β b} (h : Sigma.mk a ∘ f = Sigma.mk b ∘ g) :
a = b ∧ HEq f g := by |
rcases ‹Nonempty γ› with ⟨i⟩
obtain rfl : a = b := congr_arg Sigma.fst (congr_fun h i)
simpa [funext_iff] using h
| 168 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
section Sigma
variable {α α₁ α₂ : Type*} {β : α → Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
namespace Sigma
instance instInhabitedSigma [Inhabited α] [Inhabited (β default)] : Inhabited (Sigma β) :=
⟨⟨default, default⟩⟩
instance instDecidableEqSigma [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] :
DecidableEq (Sigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFalse n => isFalse fun h ↦ Sigma.noConfusion h fun _ e₂ ↦ n <| eq_of_heq e₂
| _, _, _, _, isFalse n => isFalse fun h ↦ Sigma.noConfusion h fun e₁ _ ↦ n e₁
-- sometimes the built-in injectivity support does not work
@[simp] -- @[nolint simpNF]
theorem mk.inj_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :
Sigma.mk a₁ b₁ = ⟨a₂, b₂⟩ ↔ a₁ = a₂ ∧ HEq b₁ b₂ :=
⟨fun h ↦ by cases h; simp,
fun ⟨h₁, h₂⟩ ↦ by subst h₁; rw [eq_of_heq h₂]⟩
#align sigma.mk.inj_iff Sigma.mk.inj_iff
@[simp]
theorem eta : ∀ x : Σa, β a, Sigma.mk x.1 x.2 = x
| ⟨_, _⟩ => rfl
#align sigma.eta Sigma.eta
#align sigma.ext Sigma.ext
theorem ext_iff {x₀ x₁ : Sigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by
cases x₀; cases x₁; exact Sigma.mk.inj_iff
#align sigma.ext_iff Sigma.ext_iff
theorem _root_.Function.eq_of_sigmaMk_comp {γ : Type*} [Nonempty γ]
{a b : α} {f : γ → β a} {g : γ → β b} (h : Sigma.mk a ∘ f = Sigma.mk b ∘ g) :
a = b ∧ HEq f g := by
rcases ‹Nonempty γ› with ⟨i⟩
obtain rfl : a = b := congr_arg Sigma.fst (congr_fun h i)
simpa [funext_iff] using h
@[ext]
theorem subtype_ext {β : Type*} {p : α → β → Prop} :
∀ {x₀ x₁ : Σa, Subtype (p a)}, x₀.fst = x₁.fst → (x₀.snd : β) = x₁.snd → x₀ = x₁
| ⟨_, _, _⟩, ⟨_, _, _⟩, rfl, rfl => rfl
#align sigma.subtype_ext Sigma.subtype_ext
theorem subtype_ext_iff {β : Type*} {p : α → β → Prop} {x₀ x₁ : Σa, Subtype (p a)} :
x₀ = x₁ ↔ x₀.fst = x₁.fst ∧ (x₀.snd : β) = x₁.snd :=
⟨fun h ↦ h ▸ ⟨rfl, rfl⟩, fun ⟨h₁, h₂⟩ ↦ subtype_ext h₁ h₂⟩
#align sigma.subtype_ext_iff Sigma.subtype_ext_iff
@[simp]
theorem «forall» {p : (Σa, β a) → Prop} : (∀ x, p x) ↔ ∀ a b, p ⟨a, b⟩ :=
⟨fun h a b ↦ h ⟨a, b⟩, fun h ⟨a, b⟩ ↦ h a b⟩
#align sigma.forall Sigma.forall
@[simp]
theorem «exists» {p : (Σa, β a) → Prop} : (∃ x, p x) ↔ ∃ a b, p ⟨a, b⟩ :=
⟨fun ⟨⟨a, b⟩, h⟩ ↦ ⟨a, b, h⟩, fun ⟨a, b, h⟩ ↦ ⟨⟨a, b⟩, h⟩⟩
#align sigma.exists Sigma.exists
lemma exists' {p : ∀ a, β a → Prop} : (∃ a b, p a b) ↔ ∃ x : Σ a, β a, p x.1 x.2 :=
(Sigma.exists (p := fun x ↦ p x.1 x.2)).symm
lemma forall' {p : ∀ a, β a → Prop} : (∀ a b, p a b) ↔ ∀ x : Σ a, β a, p x.1 x.2 :=
(Sigma.forall (p := fun x ↦ p x.1 x.2)).symm
theorem _root_.sigma_mk_injective {i : α} : Injective (@Sigma.mk α β i)
| _, _, rfl => rfl
#align sigma_mk_injective sigma_mk_injective
theorem fst_surjective [h : ∀ a, Nonempty (β a)] : Surjective (fst : (Σ a, β a) → α) := fun a ↦
let ⟨b⟩ := h a; ⟨⟨a, b⟩, rfl⟩
theorem fst_surjective_iff : Surjective (fst : (Σ a, β a) → α) ↔ ∀ a, Nonempty (β a) :=
⟨fun h a ↦ let ⟨x, hx⟩ := h a; hx ▸ ⟨x.2⟩, @fst_surjective _ _⟩
| Mathlib/Data/Sigma/Basic.lean | 120 | 122 | theorem fst_injective [h : ∀ a, Subsingleton (β a)] : Injective (fst : (Σ a, β a) → α) := by |
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (rfl : a₁ = a₂)
exact congr_arg (mk a₁) <| Subsingleton.elim _ _
| 168 |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
namespace PSigma
variable {α : Sort*} {β : α → Sort*}
def elim {γ} (f : ∀ a, β a → γ) (a : PSigma β) : γ :=
PSigma.casesOn a f
#align psigma.elim PSigma.elim
@[simp]
theorem elim_val {γ} (f : ∀ a, β a → γ) (a b) : PSigma.elim f ⟨a, b⟩ = f a b :=
rfl
#align psigma.elim_val PSigma.elim_val
instance [Inhabited α] [Inhabited (β default)] : Inhabited (PSigma β) :=
⟨⟨default, default⟩⟩
instance decidableEq [h₁ : DecidableEq α] [h₂ : ∀ a, DecidableEq (β a)] : DecidableEq (PSigma β)
| ⟨a₁, b₁⟩, ⟨a₂, b₂⟩ =>
match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with
| _, b₁, _, b₂, isTrue (Eq.refl _) =>
match b₁, b₂, h₂ _ b₁ b₂ with
| _, _, isTrue (Eq.refl _) => isTrue rfl
| _, _, isFalse n => isFalse fun h ↦ PSigma.noConfusion h fun _ e₂ ↦ n <| eq_of_heq e₂
| _, _, _, _, isFalse n => isFalse fun h ↦ PSigma.noConfusion h fun e₁ _ ↦ n e₁
-- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/porting.20data.2Esigma.2Ebasic/near/304855864
-- for an explanation of why this is currently needed. It generates `PSigma.mk.inj`.
-- This could be done elsewhere.
gen_injective_theorems% PSigma
theorem mk.inj_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :
@PSigma.mk α β a₁ b₁ = @PSigma.mk α β a₂ b₂ ↔ a₁ = a₂ ∧ HEq b₁ b₂ :=
(Iff.intro PSigma.mk.inj) fun ⟨h₁, h₂⟩ ↦
match a₁, a₂, b₁, b₂, h₁, h₂ with
| _, _, _, _, Eq.refl _, HEq.refl _ => rfl
#align psigma.mk.inj_iff PSigma.mk.inj_iff
#align psigma.ext PSigma.ext
| Mathlib/Data/Sigma/Basic.lean | 273 | 274 | theorem ext_iff {x₀ x₁ : PSigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by |
cases x₀; cases x₁; exact PSigma.mk.inj_iff
| 168 |
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
#align_import set_theory.lists from "leanprover-community/mathlib"@"497d1e06409995dd8ec95301fa8d8f3480187f4c"
variable {α : Type*}
inductive Lists'.{u} (α : Type u) : Bool → Type u
| atom : α → Lists' α false
| nil : Lists' α true
| cons' {b} : Lists' α b → Lists' α true → Lists' α true
deriving DecidableEq
#align lists' Lists'
compile_inductive% Lists'
def Lists (α : Type*) :=
Σb, Lists' α b
#align lists Lists
namespace Lists'
instance [Inhabited α] : ∀ b, Inhabited (Lists' α b)
| true => ⟨nil⟩
| false => ⟨atom default⟩
def cons : Lists α → Lists' α true → Lists' α true
| ⟨_, a⟩, l => cons' a l
#align lists'.cons Lists'.cons
@[simp]
def toList : ∀ {b}, Lists' α b → List (Lists α)
| _, atom _ => []
| _, nil => []
| _, cons' a l => ⟨_, a⟩ :: l.toList
#align lists'.to_list Lists'.toList
-- Porting note (#10618): removed @[simp]
-- simp can prove this: by simp only [@Lists'.toList, @Sigma.eta]
| Mathlib/SetTheory/Lists.lean | 88 | 88 | theorem toList_cons (a : Lists α) (l) : toList (cons a l) = a :: l.toList := by | simp
| 169 |
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