mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Ring.Subring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonAssocRing R
G : AddSubgroup R
k : ℤ
g : R
h : g ∈ G
⊢ ↑k * g ∈ G

Tactic:
convert AddSubgroup.zsmul_mem G h k

Goal state after:
R : Type u
inst✝ : NonAssocRing R
G : AddSubgroup R
k : ℤ
g : R
h : g ∈ G
⊢ ↑k * g = k • g
Module: Mathlib.Algebra.Ring.Subring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonAssocRing R
G : AddSubgroup R
k : ℤ
g : R
h : g ∈ G
⊢ ↑k * g ∈ G

Tactic:
convert AddSubgroup.zsmul_mem G h k
  rw [zsmul_eq_mul]

Goal state after:
Module: Mathlib.Algebra.Module.Submodule.Range

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
R₃ : Type u_3
M : Type u_5
M₂ : Type u_6
M₃ : Type u_7
inst✝¹⁰ : Semiring R
inst✝⁹ : Semiring R₂
inst✝⁸ : Semiring R₃
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
τ₁₂ : R →+* R₂
τ₂₃ : R₂ →+* R₃
τ₁₃ : R →+* R₃
inst✝¹ : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
g : M₂ →ₛₗ[τ₂₃] M₃
h : g ∘ₛₗ f = 0
x : M₂
hx : x ∈ f.range
y : M
hy : f y = x
⊢ g x = 0

Tactic:
rw [← hy, ← comp_apply, h, zero_apply]

Goal state after:
Module: Mathlib.Algebra.Module.Submodule.Range

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u_5
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
M' : Type u_10
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
O : Submodule R M
ϕ : ↥O →ₗ[R] M'
N : Submodule R M
x : M'
⊢ (∃ y, O.subtype y ∈ N ∧ ϕ y = x) → ∃ y, ∃ (yO : y ∈ O), y ∈ N ∧ ϕ ⟨y, yO⟩ = x

Tactic:
rintro ⟨⟨y, yO⟩, yN : y ∈ N, h⟩
    exact ⟨y, yO, yN, h⟩

Goal state after:
Module: Mathlib.Order.ModularLattice

Goal state before:
pp: α : Type u_1
a b c : α
inst✝² : Lattice α
inst✝¹ : OrderBot α
inst✝ : IsModularLattice α
h : Disjoint b c
hsup : Disjoint a (b ⊔ c)
⊢ Disjoint (a ⊔ b) c

Tactic:
disjoint_comm,

Goal state after:
α : Type u_1
a b c : α
inst✝² : Lattice α
inst✝¹ : OrderBot α
inst✝ : IsModularLattice α
h : Disjoint b c
hsup : Disjoint a (b ⊔ c)
⊢ Disjoint c (a ⊔ b)
Module: Mathlib.LinearAlgebra.Span.Defs

Goal state before:
pp: R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
r : R
x : M
hx : x ∈ s
⊢ (fun x ↦ r • x) x ∈ ↑(span R s)

Tactic:
exact smul_mem (span R s) r (subset_span hx)

Goal state after:
Module: Mathlib.Order.SupClosed

Goal state before:
pp: α : Type u_3
inst✝ : SemilatticeSup α
s : Finset α
⊢ (supClosure ↑s).Finite

Tactic:
refine ({t ∈ s.powerset | t.Nonempty}.attach.image
    fun t ↦ t.1.sup' (mem_filter.1 t.2).2 id).finite_toSet.subset ?_
  rintro _ ⟨t, ht, hts, rfl⟩
  simp only [id_eq, coe_image, mem_image, mem_coe, mem_attach, true_and, Subtype.exists,
    Finset.mem_powerset, mem_filter]
  exact ⟨t, ⟨hts, ht⟩, rfl⟩

Goal state after:
Module: Mathlib.Order.Atoms

Goal state before:
pp: α : Type u_2
inst✝¹ : SemilatticeSup α
inst✝ : OrderTop α
a b : α
ha : IsCoatom a
hb : IsCoatom b
hab : a ≠ b
⊢ Codisjoint a b

Tactic:
simp [← ha.not_le_iff_codisjoint, hb.le_iff, hab, ha.ne_top]

Goal state after:
Module: Mathlib.LinearAlgebra.Span.Defs

Goal state before:
pp: R : Type u_1
M : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
G : Type u_9
inst✝³ : Group G
inst✝² : SMul G R
inst✝¹ : MulAction G M
inst✝ : IsScalarTower G R M
g : G
x : M
⊢ R ∙ g • x = R ∙ x

Tactic:
refine le_antisymm (span_singleton_smul_le R g x) ?_

Goal state after:
R : Type u_1
M : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
G : Type u_9
inst✝³ : Group G
inst✝² : SMul G R
inst✝¹ : MulAction G M
inst✝ : IsScalarTower G R M
g : G
x : M
⊢ R ∙ x ≤ R ∙ g • x
Module: Mathlib.Order.Atoms

Goal state before:
pp: α : Type u_4
inst✝ : CompleteLattice α
⊢ IsCoatomistic α ↔ ∀ (b : α), ∃ s, b = sInf s ∧ ∀ a ∈ s, IsCoatom a

Tactic:
by
  simp_rw [isCoatomistic_iff, isGLB_iff_sInf_eq, eq_comm]

Goal state after:
Module: Mathlib.Order.SupClosed

Goal state before:
pp: case right.inf
α : Type u_3
β : Type u_4
inst✝¹ : Lattice α
inst✝ : Lattice β
s : Set α
f : α → β
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
⊢ ∀ a ∈ latticeClosure (f '' s),
    ∀ b ∈ latticeClosure (f '' s), a ∈ f '' latticeClosure s → b ∈ f '' latticeClosure s → a ⊓ b ∈ f '' latticeClosure s

Tactic:
· rintro _ - _ - ⟨a, ha, rfl⟩ ⟨b, hb, rfl⟩
    exact ⟨a ⊓ b, isSublattice_latticeClosure.infClosed ha hb, map_inf ..⟩

Goal state after:
Module: Mathlib.Order.SupIndep

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝² : Lattice α
inst✝¹ : OrderBot α
f : ι → α
inst✝ : DecidableEq ι
i j : ι
hij : i ≠ j
⊢ Disjoint (f i) (f j) → Disjoint (f j) (({j, i}.erase j).sup f)

Tactic:
simp [hij.symm, disjoint_comm]

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: case mp
α : Type u
inst✝ : CompleteLattice α
k : α
H : ∀ (s : Set α), k ≤ sSup s → ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id
⊢ ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ t.sup s

Tactic:
ι

Goal state after:
case mp
α : Type u
inst✝ : CompleteLattice α
k : α
H : ∀ (s : Set α), k ≤ sSup s → ∃ t, ↑t ⊆ s ∧ k ≤ t.sup id
ι : Type u
⊢ ∀ (s : ι → α), k ≤ iSup s → ∃ t, k ≤ t.sup s
Module: Mathlib.Order.SupIndep

Goal state before:
pp: case insert
α : Type u_1
ι : Type u_3
inst✝¹ : CompleteLattice α
inst✝ : IsModularLattice α
f : ι → α
hf : iSupIndep f
j : ι
s₀ : Finset ι
hj : j ∉ s₀
t : Set ι
hjt : j ∉ t
hst : Disjoint (↑s₀) (insert j t)
ih : Disjoint (⨆ i ∈ s₀, f i) (f j ⊔ ⨆ i ∈ t, f i)
this : Disjoint (f j) ((⨆ i ∈ t, f i) ⊔ ⨆ i ∈ ↑s₀, f i)
⊢ Disjoint (f j ⊔ ⨆ x ∈ s₀, f x) (⨆ i ∈ t, f i)

Tactic:
disjoint_comm,

Goal state after:
case insert
α : Type u_1
ι : Type u_3
inst✝¹ : CompleteLattice α
inst✝ : IsModularLattice α
f : ι → α
hf : iSupIndep f
j : ι
s₀ : Finset ι
hj : j ∉ s₀
t : Set ι
hjt : j ∉ t
hst : Disjoint (↑s₀) (insert j t)
ih : Disjoint (⨆ i ∈ s₀, f i) (f j ⊔ ⨆ i ∈ t, f i)
this : Disjoint (f j) ((⨆ i ∈ t, f i) ⊔ ⨆ i ∈ ↑s₀, f i)
⊢ Disjoint (⨆ i ∈ t, f i) (f j ⊔ ⨆ x ∈ s₀, f x)
Module: Mathlib.Algebra.Algebra.Tower

Goal state before:
pp: R : Type u
S : Type v
A : Type w
B : Type u₁
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S A
inst✝⁴ : Algebra S B
inst✝³ : Algebra R A
inst✝² : Algebra R B
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S B
f : A ≃ₐ[S] B
x : B
⊢ (restrictScalars R f).symm x = f.symm x

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Algebra.Tower

Goal state before:
pp: R : Type u
S : Type v
A : Type w
B : Type u₁
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S A
inst✝⁴ : Algebra S B
inst✝³ : Algebra R A
inst✝² : Algebra R B
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S B
f : A ≃ₐ[S] B
⊢ (restrictScalars R f).symm.toRingEquiv = f.symm.toRingEquiv

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Algebra.Tower

Goal state before:
pp: R : Type u
S : Type v
A : Type w
B : Type u₁
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra S A
inst✝⁴ : Algebra S B
inst✝³ : Algebra R A
inst✝² : Algebra R B
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S B
f : A ≃ₐ[S] B
⊢ ⇑(restrictScalars R f).symm = ⇑f.symm

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: case neg
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a : α
s : Set α
h : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
hs : s = ∅
⊢ ∀ (c : α), IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ b ∈ s, a ⊓ b

Tactic:
simp [hs]

Goal state after:
Module: Mathlib.LinearAlgebra.Span.Basic

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
ι : Sort u_8
p : ι → Submodule R M
motive : (x : M) → x ∈ ⨆ i, p i → Prop
mem : ∀ (i : ι) (x : M) (hx : x ∈ p i), motive x ⋯
zero : motive 0 ⋯
add : ∀ (x y : M) (hx : x ∈ ⨆ i, p i) (hy : y ∈ ⨆ i, p i), motive x hx → motive y hy → motive (x + y) ⋯
x✝ : M
hx✝ : x✝ ∈ ⨆ i, p i
i : ι
x : M
hx : x ∈ p i
⊢ ∃ (hx : x ∈ ⨆ i, p i), motive x hx

Tactic:
exact ⟨_, mem _ _ hx⟩

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: ι : Sort u_1
α : Type u_2
inst✝³ : CompleteLattice α
f : ι → α
inst✝² : IsModularLattice α
inst✝¹ : IsCompactlyGenerated α
inst✝ : ComplementedLattice α
b : α
⊢ sSup {a | IsAtom a ∧ a ≤ b} = b

Tactic:
have hle : sSup { a : α | IsAtom a ∧ a ≤ b } ≤ b := sSup_le fun _ => And.right

Goal state after:
ι : Sort u_1
α : Type u_2
inst✝³ : CompleteLattice α
f : ι → α
inst✝² : IsModularLattice α
inst✝¹ : IsCompactlyGenerated α
inst✝ : ComplementedLattice α
b : α
hle : sSup {a | IsAtom a ∧ a ≤ b} ≤ b
⊢ sSup {a | IsAtom a ∧ a ≤ b} = b
Module: Mathlib.Algebra.Group.Indicator

Goal state before:
pp: α : Type u_1
G : Type u_6
inst✝ : Group G
s t : Set α
h : s ⊆ t
f : α → G
⊢ (t ∪ s).mulIndicator f = t.mulIndicator f

pp: case h
α : Type u_1
G : Type u_6
inst✝ : Group G
s t : Set α
h : s ⊆ t
f : α → G
⊢ Disjoint (t \ s) s

Tactic:
union_eq_self_of_subset_right h

Goal state after:
α : Type u_1
G : Type u_6
inst✝ : Group G
s t : Set α
h : s ⊆ t
f : α → G
⊢ t.mulIndicator f = t.mulIndicator f

case h
α : Type u_1
G : Type u_6
inst✝ : Group G
s t : Set α
h : s ⊆ t
f : α → G
⊢ Disjoint (t \ s) s
Module: Mathlib.LinearAlgebra.Span.Basic

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : Semiring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
p q : Submodule R M
hpq : Disjoint p q
hker : f.ker ≤ q
⊢ Disjoint (map f p) (map f q)

Tactic:
rw [disjoint_iff, map_inf_eq_map_inf_comap, comap_map_eq, eq_bot_iff, map_le_iff_le_comap,
    comap_bot, sup_eq_left.mpr hker, hpq.eq_bot]

Goal state after:
R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : Semiring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
p q : Submodule R M
hpq : Disjoint p q
hker : f.ker ≤ q
⊢ ⊥ ≤ f.ker
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: case inr
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
b c : α
hbc : b ≤ c
h✝ : sSup {a | a ≤ c ∧ IsAtom a} = c
s : Set α
s_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a ∧ a ≤ c} → s ⊆ t → s = t
s_ind : sSupIndep s
b_inf_Sup_s : Disjoint b (sSup s)
s_atoms : ∀ a ∈ s, IsAtom a ∧ a ≤ c
a : α
ha : a ∈ {a | a ≤ c ∧ IsAtom a}
con : Disjoint a (b ⊔ sSup s)
a_dis_Sup_s : Disjoint a (sSup s)
x : α
hx : x ∈ s ∨ x = a
xa : x ≠ a
h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a}
⊢ Disjoint x (sSup (s \ {x}) ⊔ a)

Tactic:
apply
        (s_ind (hx.resolve_right xa)).disjoint_sup_right_of_disjoint_sup_left
          (a_dis_Sup_s.mono_right _).symm

Goal state after:
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
b c : α
hbc : b ≤ c
h✝ : sSup {a | a ≤ c ∧ IsAtom a} = c
s : Set α
s_max : ∀ ⦃t : Set α⦄, t ∈ {s | sSupIndep s ∧ Disjoint b (sSup s) ∧ ∀ a ∈ s, IsAtom a ∧ a ≤ c} → s ⊆ t → s = t
s_ind : sSupIndep s
b_inf_Sup_s : Disjoint b (sSup s)
s_atoms : ∀ a ∈ s, IsAtom a ∧ a ≤ c
a : α
ha : a ∈ {a | a ≤ c ∧ IsAtom a}
con : Disjoint a (b ⊔ sSup s)
a_dis_Sup_s : Disjoint a (sSup s)
x : α
hx : x ∈ s ∨ x = a
xa : x ≠ a
h : (s ∪ {a}) \ {x} = s \ {x} ∪ {a}
⊢ x ⊔ sSup (s \ {x}) ≤ sSup s
Module: Mathlib.Algebra.BigOperators.Ring.Finset

Goal state before:
pp: ι : Type u_1
R : Type u_4
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
f g : ι → R
s : Finset ι
⊢ ∀ a ∈ s.powerset, {a_1 ∈ s | ∃ (h : a_1 ∈ s), (fun a_2 x ↦ a_2 ∈ a) a_1 h} = a

Tactic:
simp_rw [Finset.ext_iff, mem_filter, mem_powerset]

Goal state after:
ι : Type u_1
R : Type u_4
inst✝¹ : CommSemiring R
inst✝ : DecidableEq ι
f g : ι → R
s : Finset ι
⊢ ∀ a ⊆ s, ∀ (a_1 : ι), (a_1 ∈ s ∧ ∃ (_ : a_1 ∈ s), a_1 ∈ a) ↔ a_1 ∈ a
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: case mpr
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
a✝ : IsAtomistic α
⊢ ComplementedLattice α

Tactic:
· exact complementedLattice_of_isAtomistic

Goal state after:
Module: Mathlib.LinearAlgebra.Span.Basic

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : Semiring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup M₂
inst✝² : Module R M
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
hf : f.range = ⊤
p : Submodule R M
⊢ map f p = ⊤ ↔ p ⊔ f.ker = ⊤

Tactic:
by
  simp_rw [← top_le_iff, ← hf, range_eq_map, LinearMap.map_le_map_iff]

Goal state after:
Module: Mathlib.LinearAlgebra.Span.Basic

Goal state before:
pp: R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
⊢ toSpanSingleton R M x = 0 ↔ x = 0

Tactic:
← toSpanSingleton_zero,

Goal state after:
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
⊢ toSpanSingleton R M x = toSpanSingleton R M 0 ↔ x = 0
Module: Mathlib.Data.Finsupp.Single

Goal state before:
pp: α : Type u_1
M : Type u_5
inst✝¹ : Zero M
inst✝ : Nonempty α
f : α →₀ M
⊢ #f.support ≤ 1 ↔ ∃ a b, f = single a b

Tactic:
by
  simp only [card_le_one_iff_subset_singleton, support_subset_singleton']

Goal state after:
Module: Mathlib.Data.Finsupp.Single

Goal state before:
pp: α : Type u_1
M : Type u_5
inst✝ : Zero M
a : α
⊢ erase a 0 = 0

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Data.Finsupp.Option

Goal state before:
pp: α : Type u_1
M : Type u_2
inst✝ : Zero M
y : M
f : α →₀ M
h : f ≠ 0
⊢ optionElim y f ≠ 0

Tactic:
contrapose h with c
  ext a
  have : f.optionElim y (Option.some a) = (0 : Option α →₀ M) (Option.some a) := by
    rw [c]
  simp only [optionElim_apply_eq_elim, Option.elim_some, coe_zero, Pi.zero_apply] at this
  exact this

Goal state after:
Module: Mathlib.Algebra.BigOperators.Finsupp.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
N : Type u_16
inst✝ : Zero N
f : α →₀ N
a : α
⊢ (if a ∈ f.support then f a else 0) = f a

Tactic:
simp only [mem_support_iff, ne_eq, ite_eq_left_iff, not_not]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
N : Type u_16
inst✝ : Zero N
f : α →₀ N
a : α
⊢ f a = 0 → 0 = f a
Module: Mathlib.Algebra.BigOperators.Finsupp.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
N : Type u_16
inst✝ : AddCommMonoid N
f : α →₀ N
a : α
⊢ (f.sum fun x v ↦ if x = a then v else 0) = f a

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.Indicator

Goal state before:
pp: ι : Type u_1
M₀ : Type u_4
inst✝¹ : MulZeroOneClass M₀
s : Set ι
i : ι
inst✝ : Nontrivial M₀
⊢ s.indicator 1 i = 0 ↔ ¬i ∈ s

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Module.Basic

Goal state before:
pp: case neg
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
x : M
hR : ↑n = 0
hS : ¬↑n = 0
⊢ ∀ (y : M), f y = 0

Tactic:
clear x

Goal state after:
case neg
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
hR : ↑n = 0
hS : ¬↑n = 0
⊢ ∀ (y : M), f y = 0
Module: Mathlib.Algebra.Module.Basic

Goal state before:
pp: case pos
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
x : M
hR : ¬↑n = 0
hS : ↑n = 0
⊢ ∀ (y : M), f y = 0

Tactic:
clear x

Goal state after:
case pos
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
hR : ¬↑n = 0
hS : ↑n = 0
⊢ ∀ (y : M), f y = 0
Module: Mathlib.Algebra.BigOperators.Finsupp.Basic

Goal state before:
pp: ι : Type u_2
A : Type u_4
B : Type u_5
inst✝¹ : AddCommMonoid A
inst✝ : AddCommMonoid B
f₁ f₂ : ι
g₁ g₂ : A
F : ι → A → B
H : f₁ ≠ f₂
HF : ∀ (f : ι), F f 0 = 0
⊢ (single f₁ g₁ + single f₂ g₂).sum F = F f₁ g₁ + F f₂ g₂

Tactic:
simp [sum_of_support_subset _ support_single_add_single_subset, single_apply, H, HF, H.symm]

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.Multiset

Goal state before:
pp: α : Type u_2
inst✝² : CommMonoid α
inst✝¹ : Preorder α
s : Multiset α
a : α
inst✝ : MulLeftMono α
h : ∀ (x : α), x ∈ s → a ≤ x
⊢ (map (Function.const α a) s).prod ≤ s.prod

Tactic:
exact prod_map_le_prod _ h

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Group.Multiset

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : CommMonoid α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedMonoid α
s : Multiset ι
f g : ι → α
⊢ (map (fun i ↦ min (f i) (g i)) s).prod ≤ min (map f s).prod (map g s).prod

Tactic:
obtain ⟨l⟩ := s
  simp_rw [Multiset.quot_mk_to_coe'', Multiset.map_coe, Multiset.prod_coe]
  apply List.prod_min_le

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Ring.Multiset

Goal state before:
pp: case mk
R : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : PartialOrder R
inst✝² : CanonicallyOrderedAdd R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
m : Multiset R
l : List R
⊢ 0 < prod (Quot.mk (⇑(List.isSetoid R)) l) ↔ ∀ (x : R), x ∈ Quot.mk (⇑(List.isSetoid R)) l → 0 < x

Tactic:
rw [Multiset.quot_mk_to_coe'', Multiset.prod_coe]

Goal state after:
case mk
R : Type u_1
inst✝⁴ : CommSemiring R
inst✝³ : PartialOrder R
inst✝² : CanonicallyOrderedAdd R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
m : Multiset R
l : List R
⊢ 0 < l.prod ↔ ∀ (x : R), x ∈ ↑l → 0 < x
Module: Mathlib.Data.Finsupp.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
M : Type u_5
inst✝ : AddCommMonoid M
f : α → β
hf : Surjective f
⊢ Surjective (mapDomain f)

Tactic:
intro x
  use mapDomain (surjInv hf) x
  rw [← mapDomain_comp, (rightInverse_surjInv hf).id, mapDomain_id]

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Ring.Multiset

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : CommMonoidWithZero β
inst✝² : PartialOrder β
inst✝¹ : PosMulMono β
inst✝ : ZeroLEOneClass β
f : α → β
h1 : ∀ (a : α), 1 ≤ f a
a : α
s' : Multiset α
ha : a ∈ a ::ₘ s'
⊢ f a * 1 ≤ f a * (map f s').prod

Tactic:
gcongr
      · exact le_trans (zero_le_one' β) (h1 a)
      · simp_all [one_le_prod]

Goal state after:
Module: Mathlib.Data.Finsupp.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M✝ : Type u_5
N : Type u_6
P✝ : Type u_7
G : Type u_8
H : Type u_9
R : Type u_10
S : Type u_11
M : Type u_12
inst✝¹ : Zero M
P : α → Prop
inst✝ : DecidablePred P
f : Subtype P →₀ M
g : { a // ¬P a } →₀ M
⊢ Disjoint (map (Embedding.subtype P) f.support) (map (Embedding.subtype fun a ↦ ¬P a) g.support)

Tactic:
simp_rw [Finset.disjoint_left, mem_map, forall_exists_index, Embedding.coe_subtype,
      Subtype.forall, Subtype.exists]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M✝ : Type u_5
N : Type u_6
P✝ : Type u_7
G : Type u_8
H : Type u_9
R : Type u_10
S : Type u_11
M : Type u_12
inst✝¹ : Zero M
P : α → Prop
inst✝ : DecidablePred P
f : Subtype P →₀ M
g : { a // ¬P a } →₀ M
⊢ ∀ ⦃a : α⦄ (a_1 : α) (b : P a_1),
    ⟨a_1, b⟩ ∈ f.support ∧ a_1 = a → ¬∃ a_2, ∃ (h : ¬P a_2), ⟨a_2, ⋯⟩ ∈ g.support ∧ a_2 = a
Module: Mathlib.Algebra.Order.BigOperators.Group.Finset

Goal state before:
pp: case cons
α : Type u_2
β : Type u_3
inst✝ : DecidableEq α
f : β → Multiset α
z : β
i : Finset β
hz : z ∉ i
hr : i.sum f = i.sup f ↔ ∀ x ∈ i, ∀ y ∈ i, x ≠ y → Disjoint (f x) (f y)
this : ∀ x ∈ i, x ≠ z
⊢ f z + ∑ x ∈ i, f x = f z ∪ i.sup f ↔ Disjoint (f z) (i.sum f) ∧ Disjoint (i.sum f) (f z) ∧ i.sum f = i.sup f

Tactic:
disjoint_comm,

Goal state after:
case cons
α : Type u_2
β : Type u_3
inst✝ : DecidableEq α
f : β → Multiset α
z : β
i : Finset β
hz : z ∉ i
hr : i.sum f = i.sup f ↔ ∀ x ∈ i, ∀ y ∈ i, x ≠ y → Disjoint (f x) (f y)
this : ∀ x ∈ i, x ≠ z
⊢ f z + ∑ x ∈ i, f x = f z ∪ i.sup f ↔ Disjoint (f z) (i.sum f) ∧ Disjoint (f z) (i.sum f) ∧ i.sum f = i.sup f
Module: Mathlib.LinearAlgebra.Finsupp.Defs

Goal state before:
pp: R✝ : Type u_1
M✝ : Type u_2
N : Type u_3
inst✝⁷ : Semiring R✝
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : Module R✝ M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R✝ N
ι : Type u_4
R : Type u_5
M : Type u_6
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i : ι
f : End (End R (ι →₀ M)) (ι →₀ M)
x : ι →₀ M
j : ι
⊢ (((↑↑(ringHomEndFinsupp ι)).toFun
          ((fun f ↦ { toFun := fun m ↦ (f (single i m)) i, map_add' := ⋯, map_smul' := ⋯ }) f))
        x)
      j =
    (f x) j

Tactic:
change f (Finsupp.lsingle (R := R) (M := M) i ∘ₗ Finsupp.lapply j • x) i = _

Goal state after:
R✝ : Type u_1
M✝ : Type u_2
N : Type u_3
inst✝⁷ : Semiring R✝
inst✝⁶ : AddCommMonoid M✝
inst✝⁵ : Module R✝ M✝
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R✝ N
ι : Type u_4
R : Type u_5
M : Type u_6
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
i : ι
f : End (End R (ι →₀ M)) (ι →₀ M)
x : ι →₀ M
j : ι
⊢ (f (lsingle i ∘ₗ lapply j • x)) i = (f x) j
Module: Mathlib.LinearAlgebra.Finsupp.Supported

Goal state before:
pp: α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
δ : Type u_7
s : δ → Set α
this✝ : DecidablePred fun x ↦ x ∈ ⋃ i, s i
this : ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)).range ≤ ⨆ i, supported M R (s i)
⊢ supported M R (⋃ i, s i) ≤ ⨆ i, supported M R (s i)

Tactic:
by
    rwa [LinearMap.range_comp, range_restrictDom, Submodule.map_top, range_subtype] at this

Goal state after:
Module: Mathlib.LinearAlgebra.Finsupp.Supported

Goal state before:
pp: case pos
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
δ : Type u_7
s : δ → Set α
this : DecidablePred fun x ↦ x ∈ ⋃ i, s i
x : α
a : M
l : α →₀ M
a✝¹ : x ∉ l.support
a✝ : a ≠ 0
ih : l ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))
h : ∃ i, x ∈ s i
⊢ single x a ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))

pp: case neg
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
δ : Type u_7
s : δ → Set α
this : DecidablePred fun x ↦ x ∈ ⋃ i, s i
x : α
a : M
l : α →₀ M
a✝¹ : x ∉ l.support
a✝ : a ≠ 0
ih : l ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))
h : ¬∃ i, x ∈ s i
⊢ single x a ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))

Tactic:
· simp only [mem_comap, coe_comp, coe_subtype, Function.comp_apply, restrictDom_apply,
        mem_iUnion, h, filter_single_of_pos]
      obtain ⟨i, hi⟩ := h
      exact le_iSup (fun i => supported M R (s i)) i (single_mem_supported R _ hi)

Goal state after:
case neg
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
δ : Type u_7
s : δ → Set α
this : DecidablePred fun x ↦ x ∈ ⋃ i, s i
x : α
a : M
l : α →₀ M
a✝¹ : x ∉ l.support
a✝ : a ≠ 0
ih : l ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))
h : ¬∃ i, x ∈ s i
⊢ single x a ∈ comap ((supported M R (⋃ i, s i)).subtype ∘ₗ restrictDom M R (⋃ i, s i)) (⨆ i, supported M R (s i))