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 Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 141 | 146 | theorem coe_prod (v : PrimeMultiset) : (v.prod : ℕ) = (v : Multiset ℕ).prod := by |
let h : (v.prod : ℕ) = ((v.map Coe.coe).map Coe.coe).prod :=
PNat.coeMonoidHom.map_multiset_prod v.toPNatMultiset
rw [Multiset.map_map] at h
have : (Coe.coe : ℕ+ → ℕ) ∘ (Coe.coe : Nat.Primes → ℕ+) = Coe.coe := funext fun p => rfl
rw [this] at h; exact h
| 837 |
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open List Function
universe u v
variable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι} {f : ∀ i, α i}
namespace List
variable (α)
abbrev TProd (l : List ι) : Type v... | Mathlib/Data/Prod/TProd.lean | 90 | 90 | theorem elim_self (v : TProd α (i :: l)) : v.elim (l.mem_cons_self i) = v.1 := by | simp [TProd.elim]
| 838 |
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open List Function
universe u v
variable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι} {f : ∀ i, α i}
namespace List
variable (α)
abbrev TProd (l : List ι) : Type v... | Mathlib/Data/Prod/TProd.lean | 94 | 95 | theorem elim_of_ne (hj : j ∈ i :: l) (hji : j ≠ i) (v : TProd α (i :: l)) :
v.elim hj = TProd.elim v.2 ((List.mem_cons.mp hj).resolve_left hji) := by | simp [TProd.elim, hji]
| 838 |
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open List Function
universe u v
variable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι} {f : ∀ i, α i}
namespace List
variable (α)
abbrev TProd (l : List ι) : Type v... | Mathlib/Data/Prod/TProd.lean | 99 | 103 | theorem elim_of_mem (hl : (i :: l).Nodup) (hj : j ∈ l) (v : TProd α (i :: l)) :
v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj := by |
apply elim_of_ne
rintro rfl
exact hl.not_mem hj
| 838 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 66 | 67 | theorem nthLe_range'_1 {n m} (i) (H : i < (range' n m).length) :
nthLe (range' n m) i H = n + i := by | simp
| 839 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 79 | 80 | theorem pairwise_lt_range (n : ℕ) : Pairwise (· < ·) (range n) := by |
simp (config := {decide := true}) only [range_eq_range', pairwise_lt_range']
| 839 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 87 | 90 | theorem take_range (m n : ℕ) : take m (range n) = range (min m n) := by |
apply List.ext_get
· simp
· simp (config := { contextual := true }) [← get_take, Nat.lt_min]
| 839 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 92 | 93 | theorem nodup_range (n : ℕ) : Nodup (range n) := by |
simp (config := {decide := true}) only [range_eq_range', nodup_range']
| 839 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 104 | 112 | theorem chain'_range_succ (r : ℕ → ℕ → Prop) (n : ℕ) :
Chain' r (range n.succ) ↔ ∀ m < n, r m m.succ := by |
rw [range_succ]
induction' n with n hn
· simp
· rw [range_succ]
simp only [append_assoc, singleton_append, chain'_append_cons_cons, chain'_singleton,
and_true_iff]
rw [hn, forall_lt_succ]
| 839 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 115 | 118 | theorem chain_range_succ (r : ℕ → ℕ → Prop) (n a : ℕ) :
Chain r a (range n.succ) ↔ r a 0 ∧ ∀ m < n, r m m.succ := by |
rw [range_succ_eq_map, chain_cons, and_congr_right_iff, ← chain'_range_succ, range_succ_eq_map]
exact fun _ => Iff.rfl
| 839 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 125 | 126 | theorem pairwise_gt_iota (n : ℕ) : Pairwise (· > ·) (iota n) := by |
simpa only [iota_eq_reverse_range', pairwise_reverse] using pairwise_lt_range' 1 n
| 839 |
import Mathlib.Data.List.Range
import Mathlib.Data.Multiset.Range
#align_import data.multiset.nodup from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
open Function List
variable {α β γ : Type*} {r : α → α → Prop} {s t : Multiset α} {a : α}
-- nodup
def Nodup (s ... | Mathlib/Data/Multiset/Nodup.lean | 96 | 100 | theorem count_eq_of_nodup [DecidableEq α] {a : α} {s : Multiset α} (d : Nodup s) :
count a s = if a ∈ s then 1 else 0 := by |
split_ifs with h
· exact count_eq_one_of_mem d h
· exact count_eq_zero_of_not_mem h
| 840 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 112 | 113 | theorem le_dedup_self {s : Multiset α} : s ≤ dedup s ↔ Nodup s := by |
rw [le_dedup, and_iff_right le_rfl]
| 841 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 116 | 117 | theorem dedup_ext {s t : Multiset α} : dedup s = dedup t ↔ ∀ a, a ∈ s ↔ a ∈ t := by |
simp [Nodup.ext]
| 841 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 120 | 122 | theorem dedup_map_dedup_eq [DecidableEq β] (f : α → β) (s : Multiset α) :
dedup (map f (dedup s)) = dedup (map f s) := by |
simp [dedup_ext]
| 841 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 126 | 128 | theorem dedup_nsmul {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : (n • s).dedup = s.dedup := by |
ext a
by_cases h : a ∈ s <;> simp [h, h0]
| 841 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 74 | 75 | theorem length_ndinsert_of_mem {a : α} {s : Multiset α} (h : a ∈ s) :
card (ndinsert a s) = card s := by | simp [h]
| 842 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 79 | 80 | theorem length_ndinsert_of_not_mem {a : α} {s : Multiset α} (h : a ∉ s) :
card (ndinsert a s) = card s + 1 := by | simp [h]
| 842 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 83 | 84 | theorem dedup_cons {a : α} {s : Multiset α} : dedup (a ::ₘ s) = ndinsert a (dedup s) := by |
by_cases h : a ∈ s <;> simp [h]
| 842 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 100 | 117 | theorem attach_ndinsert (a : α) (s : Multiset α) :
(s.ndinsert a).attach =
ndinsert ⟨a, mem_ndinsert_self a s⟩ (s.attach.map fun p => ⟨p.1, mem_ndinsert_of_mem p.2⟩) :=
have eq :
∀ h : ∀ p : { x // x ∈ s }, p.1 ∈ s,
(fun p : { x // x ∈ s } => ⟨p.val, h p⟩ : { x // x ∈ s } → { x // x ∈ s }) = id :=... |
intro t ht
by_cases h : a ∈ s
· rw [ndinsert_of_mem h] at ht
subst ht
rw [eq, map_id, ndinsert_of_mem (mem_attach _ _)]
· rw [ndinsert_of_not_mem h] at ht
subst ht
simp [attach_cons, h]
this _ rfl
| 842 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 127 | 129 | theorem disjoint_ndinsert_right {a : α} {s t : Multiset α} :
Disjoint s (ndinsert a t) ↔ a ∉ s ∧ Disjoint s t := by |
rw [disjoint_comm, disjoint_ndinsert_left]; tauto
| 842 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 82 | 86 | theorem map_join (f : α → β) (S : Multiset (Multiset α)) :
map f (join S) = join (map (map f) S) := by |
induction S using Multiset.induction with
| empty => simp
| cons _ _ ih => simp [ih]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 89 | 93 | theorem prod_join [CommMonoid α] {S : Multiset (Multiset α)} :
prod (join S) = prod (map prod S) := by |
induction S using Multiset.induction with
| empty => simp
| cons _ _ ih => simp [ih]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 95 | 98 | theorem rel_join {r : α → β → Prop} {s t} (h : Rel (Rel r) s t) : Rel r s.join t.join := by |
induction h with
| zero => simp
| cons hab hst ih => simpa using hab.add ih
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 115 | 117 | theorem coe_bind (l : List α) (f : α → List β) : (@bind α β l fun a => f a) = l.bind f := by |
rw [List.bind, ← coe_join, List.map_map]
rfl
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 126 | 126 | theorem cons_bind : (a ::ₘ s).bind f = f a + s.bind f := by | simp [bind]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 130 | 130 | theorem singleton_bind : bind {a} f = f a := by | simp [bind]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 134 | 134 | theorem add_bind : (s + t).bind f = s.bind f + t.bind f := by | simp [bind]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 138 | 138 | theorem bind_zero : s.bind (fun _ => 0 : α → Multiset β) = 0 := by | simp [bind, join, nsmul_zero]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 142 | 142 | theorem bind_add : (s.bind fun a => f a + g a) = s.bind f + s.bind g := by | simp [bind, join]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 158 | 159 | theorem mem_bind {b s} {f : α → Multiset β} : b ∈ bind s f ↔ ∃ a ∈ s, b ∈ f a := by |
simp [bind]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 163 | 163 | theorem card_bind : card (s.bind f) = (s.map (card ∘ f)).sum := by | simp [bind]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 166 | 167 | theorem bind_congr {f g : α → Multiset β} {m : Multiset α} :
(∀ a ∈ m, f a = g a) → bind m f = bind m g := by | simp (config := { contextual := true }) [bind]
| 843 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 170 | 174 | theorem bind_hcongr {β' : Type v} {m : Multiset α} {f : α → Multiset β} {f' : α → Multiset β'}
(h : β = β') (hf : ∀ a ∈ m, HEq (f a) (f' a)) : HEq (bind m f) (bind m f') := by |
subst h
simp only [heq_eq_eq] at hf
simp [bind_congr hf]
| 843 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.sections from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists Ring
namespace Multiset
variable {α : Type*}
section Sections
def Sections (s : Multiset (Multiset α)) : Multiset (Multiset α) :=
Multiset.... | Mathlib/Data/Multiset/Sections.lean | 60 | 64 | theorem mem_sections {s : Multiset (Multiset α)} :
∀ {a}, a ∈ Sections s ↔ s.Rel (fun s a => a ∈ s) a := by |
induction s using Multiset.induction_on with
| empty => simp
| cons _ _ ih => simp [ih, rel_cons_left, eq_comm]
| 844 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 63 | 64 | theorem fold_cons_right (b a : α) (s : Multiset α) : (a ::ₘ s).fold op b = s.fold op b * a := by |
simp [hc.comm]
| 845 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 67 | 68 | theorem fold_cons'_right (b a : α) (s : Multiset α) : (a ::ₘ s).fold op b = s.fold op (b * a) := by |
rw [fold_eq_foldl, foldl_cons, ← fold_eq_foldl]
| 845 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 71 | 72 | theorem fold_cons'_left (b a : α) (s : Multiset α) : (a ::ₘ s).fold op b = s.fold op (a * b) := by |
rw [fold_cons'_right, hc.comm]
| 845 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 82 | 87 | theorem fold_bind {ι : Type*} (s : Multiset ι) (t : ι → Multiset α) (b : ι → α) (b₀ : α) :
(s.bind t).fold op ((s.map b).fold op b₀) =
(s.map fun i => (t i).fold op (b i)).fold op b₀ := by |
induction' s using Multiset.induction_on with a ha ih
· rw [zero_bind, map_zero, map_zero, fold_zero]
· rw [cons_bind, map_cons, map_cons, fold_cons_left, fold_cons_left, fold_add, ih]
| 845 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 108 | 110 | theorem fold_union_inter [DecidableEq α] (s₁ s₂ : Multiset α) (b₁ b₂ : α) :
((s₁ ∪ s₂).fold op b₁ * (s₁ ∩ s₂).fold op b₂) = s₁.fold op b₁ * s₂.fold op b₂ := by |
rw [← fold_add op, union_add_inter, fold_add op]
| 845 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 104 | 106 | theorem lcm_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).lcm = GCDMonoid.lcm s₁.lcm s₂.lcm := by |
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add]
simp
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 110 | 112 | theorem lcm_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).lcm = GCDMonoid.lcm s₁.lcm s₂.lcm := by |
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add]
simp
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 116 | 118 | theorem lcm_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).lcm = GCDMonoid.lcm a s.lcm := by |
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_cons]
simp
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 173 | 182 | theorem gcd_eq_zero_iff (s : Multiset α) : s.gcd = 0 ↔ ∀ x : α, x ∈ s → x = 0 := by |
constructor
· intro h x hx
apply eq_zero_of_zero_dvd
rw [← h]
apply gcd_dvd hx
· refine s.induction_on ?_ ?_
· simp
intro a s sgcd h
simp [h a (mem_cons_self a s), sgcd fun x hx ↦ h x (mem_cons_of_mem hx)]
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 185 | 190 | theorem gcd_map_mul (a : α) (s : Multiset α) : (s.map (a * ·)).gcd = normalize a * s.gcd := by |
refine s.induction_on ?_ fun b s ih ↦ ?_
· simp_rw [map_zero, gcd_zero, mul_zero]
· simp_rw [map_cons, gcd_cons, ← gcd_mul_left]
rw [ih]
apply ((normalize_associated a).mul_right _).gcd_eq_right
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 207 | 209 | theorem gcd_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).gcd = GCDMonoid.gcd s₁.gcd s₂.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add]
simp
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 213 | 215 | theorem gcd_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).gcd = GCDMonoid.gcd s₁.gcd s₂.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add]
simp
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 219 | 221 | theorem gcd_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).gcd = GCDMonoid.gcd a s.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_cons]
simp
| 846 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 240 | 254 | theorem extract_gcd (s : Multiset α) (hs : s ≠ 0) :
∃ t : Multiset α, s = t.map (s.gcd * ·) ∧ t.gcd = 1 := by |
classical
by_cases h : ∀ x ∈ s, x = (0 : α)
· use replicate (card s) 1
rw [map_replicate, eq_replicate, mul_one, s.gcd_eq_zero_iff.2 h, ← nsmul_singleton,
← gcd_dedup, dedup_nsmul (card_pos.2 hs).ne', dedup_singleton, gcd_singleton]
exact ⟨⟨rfl, h⟩, normalize_one⟩
· choose f hf using @gcd... | 846 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 50 | 52 | theorem fold_cons (h : a ∉ s) : (cons a s h).fold op b f = f a * s.fold op b f := by |
dsimp only [fold]
rw [cons_val, Multiset.map_cons, fold_cons_left]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 56 | 59 | theorem fold_insert [DecidableEq α] (h : a ∉ s) :
(insert a s).fold op b f = f a * s.fold op b f := by |
unfold fold
rw [insert_val, ndinsert_of_not_mem h, Multiset.map_cons, fold_cons_left]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 68 | 69 | theorem fold_map {g : γ ↪ α} {s : Finset γ} : (s.map g).fold op b f = s.fold op b (f ∘ g) := by |
simp only [fold, map, Multiset.map_map]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 73 | 75 | theorem fold_image [DecidableEq α] {g : γ → α} {s : Finset γ}
(H : ∀ x ∈ s, ∀ y ∈ s, g x = g y → x = y) : (s.image g).fold op b f = s.fold op b (f ∘ g) := by |
simp only [fold, image_val_of_injOn H, Multiset.map_map]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 79 | 80 | theorem fold_congr {g : α → β} (H : ∀ x ∈ s, f x = g x) : s.fold op b f = s.fold op b g := by |
rw [fold, fold, map_congr rfl H]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 83 | 85 | theorem fold_op_distrib {f g : α → β} {b₁ b₂ : β} :
(s.fold op (b₁ * b₂) fun x => f x * g x) = s.fold op b₁ f * s.fold op b₂ g := by |
simp only [fold, fold_distrib]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 88 | 96 | theorem fold_const [hd : Decidable (s = ∅)] (c : β) (h : op c (op b c) = op b c) :
Finset.fold op b (fun _ => c) s = if s = ∅ then b else op b c := by |
classical
induction' s using Finset.induction_on with x s hx IH generalizing hd
· simp
· simp only [Finset.fold_insert hx, IH, if_false, Finset.insert_ne_empty]
split_ifs
· rw [hc.comm]
· exact h
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 99 | 103 | theorem fold_hom {op' : γ → γ → γ} [Std.Commutative op'] [Std.Associative op'] {m : β → γ}
(hm : ∀ x y, m (op x y) = op' (m x) (m y)) :
(s.fold op' (m b) fun x => m (f x)) = m (s.fold op b f) := by |
rw [fold, fold, ← Multiset.fold_hom op hm, Multiset.map_map]
simp only [Function.comp_apply]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 116 | 120 | theorem fold_union_inter [DecidableEq α] {s₁ s₂ : Finset α} {b₁ b₂ : β} :
((s₁ ∪ s₂).fold op b₁ f * (s₁ ∩ s₂).fold op b₂ f) = s₁.fold op b₂ f * s₂.fold op b₁ f := by |
unfold fold
rw [← fold_add op, ← Multiset.map_add, union_val, inter_val, union_add_inter, Multiset.map_add,
hc.comm, fold_add]
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 124 | 129 | theorem fold_insert_idem [DecidableEq α] [hi : Std.IdempotentOp op] :
(insert a s).fold op b f = f a * s.fold op b f := by |
by_cases h : a ∈ s
· rw [← insert_erase h]
simp [← ha.assoc, hi.idempotent]
· apply fold_insert h
| 847 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 132 | 138 | theorem fold_image_idem [DecidableEq α] {g : γ → α} {s : Finset γ} [hi : Std.IdempotentOp op] :
(image g s).fold op b f = s.fold op b (f ∘ g) := by |
induction' s using Finset.cons_induction with x xs hx ih
· rw [fold_empty, image_empty, fold_empty]
· haveI := Classical.decEq γ
rw [fold_cons, cons_eq_insert, image_insert, fold_insert_idem, ih]
simp only [Function.comp_apply]
| 847 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 62 | 65 | theorem lcm_dvd_iff {a : α} : s.lcm f ∣ a ↔ ∀ b ∈ s, f b ∣ a := by |
apply Iff.trans Multiset.lcm_dvd
simp only [Multiset.mem_map, and_imp, exists_imp]
exact ⟨fun k b hb ↦ k _ _ hb rfl, fun k a' b hb h ↦ h ▸ k _ hb⟩
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 77 | 82 | theorem lcm_insert [DecidableEq β] {b : β} :
(insert b s : Finset β).lcm f = GCDMonoid.lcm (f b) (s.lcm f) := by |
by_cases h : b ∈ s
· rw [insert_eq_of_mem h,
(lcm_eq_right_iff (f b) (s.lcm f) (Multiset.normalize_lcm (s.1.map f))).2 (dvd_lcm h)]
apply fold_insert h
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 92 | 92 | theorem normalize_lcm : normalize (s.lcm f) = s.lcm f := by | simp [lcm_def]
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 | theorem lcm_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.lcm f = s₂.lcm g := by |
subst hs
exact Finset.fold_congr hfg
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 114 | 116 | theorem lcm_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).lcm f = s.lcm (f ∘ g) := by |
classical induction' s using Finset.induction with c s _ ih <;> simp [*]
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 123 | 125 | theorem lcm_eq_zero_iff [Nontrivial α] : s.lcm f = 0 ↔ 0 ∈ f '' s := by |
simp only [Multiset.mem_map, lcm_def, Multiset.lcm_eq_zero_iff, Set.mem_image, mem_coe, ←
Finset.mem_def]
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 151 | 154 | theorem dvd_gcd_iff {a : α} : a ∣ s.gcd f ↔ ∀ b ∈ s, a ∣ f b := by |
apply Iff.trans Multiset.dvd_gcd
simp only [Multiset.mem_map, and_imp, exists_imp]
exact ⟨fun k b hb ↦ k _ _ hb rfl, fun k a' b hb h ↦ h ▸ k _ hb⟩
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 166 | 171 | theorem gcd_insert [DecidableEq β] {b : β} :
(insert b s : Finset β).gcd f = GCDMonoid.gcd (f b) (s.gcd f) := by |
by_cases h : b ∈ s
· rw [insert_eq_of_mem h,
(gcd_eq_right_iff (f b) (s.gcd f) (Multiset.normalize_gcd (s.1.map f))).2 (gcd_dvd h)]
apply fold_insert h
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 181 | 181 | theorem normalize_gcd : normalize (s.gcd f) = s.gcd f := by | simp [gcd_def]
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 189 | 192 | theorem gcd_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.gcd f = s₂.gcd g := by |
subst hs
exact Finset.fold_congr hfg
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 203 | 205 | theorem gcd_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).gcd f = s.gcd (f ∘ g) := by |
classical induction' s using Finset.induction with c s _ ih <;> simp [*]
| 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 212 | 223 | theorem gcd_eq_zero_iff : s.gcd f = 0 ↔ ∀ x : β, x ∈ s → f x = 0 := by |
rw [gcd_def, Multiset.gcd_eq_zero_iff]
constructor <;> intro h
· intro b bs
apply h (f b)
simp only [Multiset.mem_map, mem_def.1 bs]
use b
simp only [mem_def.1 bs, eq_self_iff_true, and_self]
· intro a as
rw [Multiset.mem_map] at as
rcases as with ⟨b, ⟨bs, rfl⟩⟩
apply h b (mem_def.1... | 848 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 228 | 241 | theorem gcd_eq_gcd_filter_ne_zero [DecidablePred fun x : β ↦ f x = 0] :
s.gcd f = (s.filter fun x ↦ f x ≠ 0).gcd f := by |
classical
trans ((s.filter fun x ↦ f x = 0) ∪ s.filter fun x ↦ (f x ≠ 0)).gcd f
· rw [filter_union_filter_neg_eq]
rw [gcd_union]
refine Eq.trans (?_ : _ = GCDMonoid.gcd (0 : α) ?_) (?_ : GCDMonoid.gcd (0 : α) _ = _)
· exact (gcd (filter (fun x => (f x ≠ 0)) s) f)
· refine congr (congr rfl <| ... | 848 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 79 | 80 | theorem sup_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
| 849 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 84 | 85 | theorem sup_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
| 849 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 89 | 90 | theorem sup_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).sup = a ⊔ s.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_cons]; simp
| 849 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 93 | 99 | theorem nodup_sup_iff {α : Type*} [DecidableEq α] {m : Multiset (Multiset α)} :
m.sup.Nodup ↔ ∀ a : Multiset α, a ∈ m → a.Nodup := by |
-- Porting note: this was originally `apply m.induction_on`, which failed due to
-- `failed to elaborate eliminator, expected type is not available`
induction' m using Multiset.induction_on with _ _ h
· simp
· simp [h]
| 849 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 163 | 164 | theorem inf_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).inf = s₁.inf ⊓ s₂.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| 849 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 168 | 169 | theorem inf_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).inf = s₁.inf ⊓ s₂.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| 849 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 173 | 174 | theorem inf_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).inf = a ⊓ s.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_cons]; simp
| 849 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Multiset
section Pi
variable {α : Type*}
open Function
def Pi.empty (δ : α → Sort*) : ∀ a ∈ (0 : Multiset α), δ a :=
nofun
#align multiset.pi.empty Multi... | Mathlib/Data/Multiset/Pi.lean | 49 | 58 | theorem Pi.cons_swap {a a' : α} {b : δ a} {b' : δ a'} {m : Multiset α} {f : ∀ a ∈ m, δ a}
(h : a ≠ a') : HEq (Pi.cons (a' ::ₘ m) a b (Pi.cons m a' b' f))
(Pi.cons (a ::ₘ m) a' b' (Pi.cons m a b f)) := by |
apply hfunext rfl
simp only [heq_iff_eq]
rintro a'' _ rfl
refine hfunext (by rw [Multiset.cons_swap]) fun ha₁ ha₂ _ => ?_
rcases ne_or_eq a'' a with (h₁ | rfl)
on_goal 1 => rcases eq_or_ne a'' a' with (rfl | h₂)
all_goals simp [*, Pi.cons_same, Pi.cons_ne]
| 850 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Multiset
section Pi
variable {α : Type*}
open Function
def Pi.empty (δ : α → Sort*) : ∀ a ∈ (0 : Multiset α), δ a :=
nofun
#align multiset.pi.empty Multi... | Mathlib/Data/Multiset/Pi.lean | 62 | 68 | theorem pi.cons_eta {m : Multiset α} {a : α} (f : ∀ a' ∈ a ::ₘ m, δ a') :
(Pi.cons m a (f _ (mem_cons_self _ _)) fun a' ha' => f a' (mem_cons_of_mem ha')) = f := by |
ext a' h'
by_cases h : a' = a
· subst h
rw [Pi.cons_same]
· rw [Pi.cons_ne _ h]
| 850 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Multiset
section Pi
variable {α : Type*}
open Function
def Pi.empty (δ : α → Sort*) : ∀ a ∈ (0 : Multiset α), δ a :=
nofun
#align multiset.pi.empty Multi... | Mathlib/Data/Multiset/Pi.lean | 71 | 80 | theorem Pi.cons_injective {a : α} {b : δ a} {s : Multiset α} (hs : a ∉ s) :
Function.Injective (Pi.cons s a b) := fun f₁ f₂ eq =>
funext fun a' =>
funext fun h' =>
have ne : a ≠ a' := fun h => hs <| h.symm ▸ h'
have : a' ∈ a ::ₘ s := mem_cons_of_mem h'
calc
f₁ a' h' = Pi.cons s a b f... | rw [Pi.cons_ne this ne.symm]
_ = Pi.cons s a b f₂ a' this := by rw [eq]
_ = f₂ a' h' := by rw [Pi.cons_ne this ne.symm]
| 850 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Pi
#align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Finset
open Multiset
section Pi
variable {α : Type*}
def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=... | Mathlib/Data/Finset/Pi.lean | 74 | 83 | theorem Pi.cons_injective {a : α} {b : δ a} {s : Finset α} (hs : a ∉ s) :
Function.Injective (Pi.cons s a b) := fun e₁ e₂ eq =>
@Multiset.Pi.cons_injective α _ δ a b s.1 hs _ _ <|
funext fun e =>
funext fun h =>
have :
Pi.cons s a b e₁ e (by simpa only [Multiset.mem_cons, mem_insert] u... |
rw [eq]
this
| 851 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Pi
#align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Finset
open Multiset
section Pi
variable {α : Type*}
def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=... | Mathlib/Data/Finset/Pi.lean | 96 | 112 | theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
(ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) := by |
apply eq_of_veq
rw [← (pi (insert a s) t).2.dedup]
refine
(fun s' (h : s' = a ::ₘ s.1) =>
(?_ :
dedup (Multiset.pi s' fun a => (t a).1) =
dedup
((t a).1.bind fun b =>
dedup <|
(Multiset.pi s.1 fun a : α => (t a).val).map fun f a' h... | 851 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Pi
#align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Finset
open Multiset
section Pi
variable {α : Type*}
def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=... | Mathlib/Data/Finset/Pi.lean | 115 | 123 | theorem pi_singletons {β : Type*} (s : Finset α) (f : α → β) :
(s.pi fun a => ({f a} : Finset β)) = {fun a _ => f a} := by |
rw [eq_singleton_iff_unique_mem]
constructor
· simp
intro a ha
ext i hi
rw [mem_pi] at ha
simpa using ha i hi
| 851 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 82 | 87 | theorem sup_sup : s.sup (f ⊔ g) = s.sup f ⊔ s.sup g := by |
induction s using Finset.cons_induction with
| empty => rw [sup_empty, sup_empty, sup_empty, bot_sup_eq]
| cons _ _ _ ih =>
rw [sup_cons, sup_cons, sup_cons, ih]
exact sup_sup_sup_comm _ _ _ _
| 852 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 90 | 93 | theorem sup_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.sup f = s₂.sup g := by |
subst hs
exact Finset.fold_congr hfg
| 852 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 140 | 143 | theorem sup_bot (s : Finset β) : (s.sup fun _ => ⊥) = (⊥ : α) := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· exact sup_empty
· exact sup_const hs _
| 852 |
import Mathlib.Order.Ideal
import Mathlib.Data.Finset.Lattice
#align_import order.countable_dense_linear_order from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open scoped Classical
namespace Order
theorem exists_between_finsets {α : Type*} [LinearOrder α] [... | Mathlib/Order/CountableDenseLinearOrder.lean | 94 | 122 | theorem exists_across [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β]
(f : PartialIso α β) (a : α) :
∃ b : β, ∀ p ∈ f.val, cmp (Prod.fst p) a = cmp (Prod.snd p) b := by |
by_cases h : ∃ b, (a, b) ∈ f.val
· cases' h with b hb
exact ⟨b, fun p hp ↦ f.prop _ hp _ hb⟩
have :
∀ x ∈ (f.val.filter fun p : α × β ↦ p.fst < a).image Prod.snd,
∀ y ∈ (f.val.filter fun p : α × β ↦ a < p.fst).image Prod.snd, x < y := by
intro x hx y hy
rw [Finset.mem_image] at hx hy
rc... | 853 |
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {α ι ι' : Type*}
instance [DecidableEq α] {r : α → α → Prop} [DecidableRel r] {s : Finset α} :
Decidable ((s : Set α).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 27 | 30 | theorem Finset.pairwiseDisjoint_range_singleton :
(Set.range (singleton : α → Finset α)).PairwiseDisjoint id := by |
rintro _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ h
exact disjoint_singleton.2 (ne_of_apply_ne _ h)
| 854 |
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {α ι ι' : Type*}
instance [DecidableEq α] {r : α → α → Prop} [DecidableRel r] {s : Finset α} :
Decidable ((s : Set α).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 44 | 48 | theorem PairwiseDisjoint.image_finset_of_le [DecidableEq ι] {s : Finset ι} {f : ι → α}
(hs : (s : Set ι).PairwiseDisjoint f) {g : ι → ι} (hf : ∀ a, f (g a) ≤ f a) :
(s.image g : Set ι).PairwiseDisjoint f := by |
rw [coe_image]
exact hs.image_of_le hf
| 854 |
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {α ι ι' : Type*}
instance [DecidableEq α] {r : α → α → Prop} [DecidableRel r] {s : Finset α} :
Decidable ((s : Set α).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 62 | 71 | theorem PairwiseDisjoint.biUnion_finset {s : Set ι'} {g : ι' → Finset ι} {f : ι → α}
(hs : s.PairwiseDisjoint fun i' : ι' => (g i').sup f)
(hg : ∀ i ∈ s, (g i : Set ι).PairwiseDisjoint f) : (⋃ i ∈ s, ↑(g i)).PairwiseDisjoint f := by |
rintro a ha b hb hab
simp_rw [Set.mem_iUnion] at ha hb
obtain ⟨c, hc, ha⟩ := ha
obtain ⟨d, hd, hb⟩ := hb
obtain hcd | hcd := eq_or_ne (g c) (g d)
· exact hg d hd (by rwa [hcd] at ha) hb hab
· exact (hs hc hd (ne_of_apply_ne _ hcd)).mono (Finset.le_sup ha) (Finset.le_sup hb)
| 854 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
| Mathlib/Data/Finsupp/BigOperators.lean | 39 | 45 | theorem List.support_sum_subset [AddMonoid M] (l : List (ι →₀ M)) :
l.sum.support ⊆ l.foldr (Finsupp.support · ⊔ ·) ∅ := by |
induction' l with hd tl IH
· simp
· simp only [List.sum_cons, Finset.union_comm]
refine Finsupp.support_add.trans (Finset.union_subset_union ?_ IH)
rfl
| 855 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 48 | 52 | theorem Multiset.support_sum_subset [AddCommMonoid M] (s : Multiset (ι →₀ M)) :
s.sum.support ⊆ (s.map Finsupp.support).sup := by |
induction s using Quot.inductionOn
simpa only [Multiset.quot_mk_to_coe'', Multiset.sum_coe, Multiset.map_coe, Multiset.sup_coe,
List.foldr_map] using List.support_sum_subset _
| 855 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 55 | 57 | theorem Finset.support_sum_subset [AddCommMonoid M] (s : Finset (ι →₀ M)) :
(s.sum id).support ⊆ Finset.sup s Finsupp.support := by |
classical convert Multiset.support_sum_subset s.1; simp
| 855 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 60 | 66 | theorem List.mem_foldr_sup_support_iff [Zero M] {l : List (ι →₀ M)} {x : ι} :
x ∈ l.foldr (Finsupp.support · ⊔ ·) ∅ ↔ ∃ f ∈ l, x ∈ f.support := by |
simp only [Finset.sup_eq_union, List.foldr_map, Finsupp.mem_support_iff, exists_prop]
induction' l with hd tl IH
· simp
· simp only [foldr, Function.comp_apply, Finset.mem_union, Finsupp.mem_support_iff, ne_eq, IH,
find?, mem_cons, exists_eq_or_imp]
| 855 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 81 | 96 | theorem List.support_sum_eq [AddMonoid M] (l : List (ι →₀ M))
(hl : l.Pairwise (_root_.Disjoint on Finsupp.support)) :
l.sum.support = l.foldr (Finsupp.support · ⊔ ·) ∅ := by |
induction' l with hd tl IH
· simp
· simp only [List.pairwise_cons] at hl
simp only [List.sum_cons, List.foldr_cons, Function.comp_apply]
rw [Finsupp.support_add_eq, IH hl.right, Finset.sup_eq_union]
suffices _root_.Disjoint hd.support (tl.foldr (fun x y ↦ (Finsupp.support x ⊔ y)) ∅) by
exact Fi... | 855 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 99 | 111 | theorem Multiset.support_sum_eq [AddCommMonoid M] (s : Multiset (ι →₀ M))
(hs : s.Pairwise (_root_.Disjoint on Finsupp.support)) :
s.sum.support = (s.map Finsupp.support).sup := by |
induction' s using Quot.inductionOn with a
obtain ⟨l, hl, hd⟩ := hs
suffices a.Pairwise (_root_.Disjoint on Finsupp.support) by
convert List.support_sum_eq a this
· simp only [Multiset.quot_mk_to_coe'', Multiset.sum_coe]
· dsimp only [Function.comp_def]
simp only [quot_mk_to_coe'', map_coe, sup... | 855 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 114 | 128 | theorem Finset.support_sum_eq [AddCommMonoid M] (s : Finset (ι →₀ M))
(hs : (s : Set (ι →₀ M)).PairwiseDisjoint Finsupp.support) :
(s.sum id).support = Finset.sup s Finsupp.support := by |
classical
suffices s.1.Pairwise (_root_.Disjoint on Finsupp.support) by
convert Multiset.support_sum_eq s.1 this
exact (Finset.sum_val _).symm
obtain ⟨l, hl, hn⟩ : ∃ l : List (ι →₀ M), l.toFinset = s ∧ l.Nodup := by
refine ⟨s.toList, ?_, Finset.nodup_toList _⟩
simp
subst hl
rwa [List.toFinset... | 855 |
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