mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Group.ULift

Goal state before:
pp: α : Type u
β : Type v
x y : ULift α
inst✝ : MulOneClass α
⊢ ∀ (x y : ULift α), Equiv.ulift (x * y) = Equiv.ulift x * Equiv.ulift y

Tactic:
intros

Goal state after:
α : Type u
β : Type v
x y : ULift α
inst✝ : MulOneClass α
x✝ y✝ : ULift α
⊢ Equiv.ulift (x✝ * y✝) = Equiv.ulift x✝ * Equiv.ulift y✝
Module: Mathlib.Algebra.Module.Submodule.Map

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p✝ p' : Submodule R M
q q' : Submodule R₂ M₂
x : M
σ₂₁ : R₂ →+* R
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : M →ₛₗ[σ₁₂] M₂
i : Injective ⇑f
p : Submodule R M
⊢ ∀ (x y : ↥p),
    (Equiv.Set.image (⇑f) (↑p) i).toFun (x + y) =
      (Equiv.Set.image (⇑f) (↑p) i).toFun x + (Equiv.Set.image (⇑f) (↑p) i).toFun y

Tactic:
intros

Goal state after:
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p✝ p' : Submodule R M
q q' : Submodule R₂ M₂
x : M
σ₂₁ : R₂ →+* R
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : M →ₛₗ[σ₁₂] M₂
i : Injective ⇑f
p : Submodule R M
x✝ y✝ : ↥p
⊢ (Equiv.Set.image (⇑f) (↑p) i).toFun (x✝ + y✝) =
    (Equiv.Set.image (⇑f) (↑p) i).toFun x✝ + (Equiv.Set.image (⇑f) (↑p) i).toFun y✝
Module: Mathlib.Algebra.Module.Submodule.Map

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p✝ p' : Submodule R M
q q' : Submodule R₂ M₂
x : M
σ₂₁ : R₂ →+* R
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : M →ₛₗ[σ₁₂] M₂
i : Injective ⇑f
p : Submodule R M
⊢ ∀ (m : R) (x : ↥p), (Equiv.Set.image (⇑f) (↑p) i).toFun (m • x) = σ₁₂ m • (Equiv.Set.image (⇑f) (↑p) i).toFun x

Tactic:
intros

Goal state after:
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
inst✝¹¹ : Semiring R
inst✝¹⁰ : Semiring R₂
inst✝⁹ : Semiring R₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : AddCommMonoid M₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
p✝ p' : Submodule R M
q q' : Submodule R₂ M₂
x : M
σ₂₁ : R₂ →+* R
inst✝¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝ : RingHomInvPair σ₂₁ σ₁₂
f : M →ₛₗ[σ₁₂] M₂
i : Injective ⇑f
p : Submodule R M
m✝ : R
x✝ : ↥p
⊢ (Equiv.Set.image (⇑f) (↑p) i).toFun (m✝ • x✝) = σ₁₂ m✝ • (Equiv.Set.image (⇑f) (↑p) i).toFun x✝
Module: Mathlib.Algebra.Module.Submodule.Map

Goal state before:
pp: R : Type u_1
R₂ : Type u_3
M : Type u_5
M₂ : Type u_7
inst✝⁶ : Semiring R
inst✝⁵ : Semiring R₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M
inst✝¹ : Module R₂ M₂
σ₁₂ : R →+* R₂
inst✝ : RingHomSurjective σ₁₂
f : M →ₛₗ[σ₁₂] M₂
hf : Injective ⇑f
p q : Submodule R M
hpq : Disjoint p q
⊢ Disjoint (map f p) (map f q)

Tactic:
disjoint_iff,

Goal state after:
R : Type u_1
R₂ : Type u_3
M : Type u_5
M₂ : Type u_7
inst✝⁶ : Semiring R
inst✝⁵ : Semiring R₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M₂
inst✝² : Module R M
inst✝¹ : Module R₂ M₂
σ₁₂ : R →+* R₂
inst✝ : RingHomSurjective σ₁₂
f : M →ₛₗ[σ₁₂] M₂
hf : Injective ⇑f
p q : Submodule R M
hpq : Disjoint p q
⊢ map f p ⊓ map f q = ⊥
Module: Mathlib.Algebra.GroupWithZero.ULift

Goal state before:
pp: α : Type u
inst✝ : MulZeroOneClass α
⊢ ∀ (a b : ULift α), Equiv.ulift (a * b) = Equiv.ulift a * Equiv.ulift b

Tactic:
intros

Goal state after:
α : Type u
inst✝ : MulZeroOneClass α
a✝ b✝ : ULift α
⊢ Equiv.ulift (a✝ * b✝) = Equiv.ulift a✝ * Equiv.ulift b✝
Module: Mathlib.RingTheory.NonUnitalSubsemiring.Basic

Goal state before:
pp: R : Type u
inst✝ : NonUnitalNonAssocSemiring R
s : Set R
p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop
mem_mem : ∀ (x : R) (hx : x ∈ s) (y : R) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯
add_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : x ∈ closure s
hy : y ∈ closure s
⊢ p x y hx hy

Tactic:
by
  induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ h _ hz
    | zero => exact zero_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
  | zero => exact zero_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂

Goal state after:
Module: Mathlib.RingTheory.NonUnitalSubring.Basic

Goal state before:
pp: case mem
R : Type u
inst✝ : NonUnitalNonAssocRing R
s : Set R
p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯
neg_left : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p (-x) y ⋯ hy
neg_right : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : x ∈ closure s
z : R
hz : z ∈ s
⊢ p x z hx ⋯

Tactic:
induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h

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

Goal state before:
pp: R : Type u
A : Type v
B : Type w
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Semiring B
inst✝ : Algebra R B
x✝ : A →ₐ[R] A
⊢ x✝ • 1 = 1

Tactic:
by ext; exact map_one _

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

Goal state before:
pp: case mp
R : Type u
A : Type v
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
M : Submonoid R
B : Type w
inst✝¹ : Semiring B
inst✝ : Algebra R B
f : A →ₐ[R] B
x : B
⊢ x ∈ Submonoid.map f (algebraMapSubmonoid A M) → x ∈ algebraMapSubmonoid B M

Tactic:
rintro ⟨a, ⟨r, hr, rfl⟩, rfl⟩
    simp only [AlgHom.commutes]
    use r

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

Goal state before:
pp: case mem
R : Type u
inst✝ : NonAssocRing R
s : Set R
p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : x ∈ closure s), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : x ∈ closure s), p x 1 hx ⋯
neg_left : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p (-x) y ⋯ hy
neg_right : ∀ (x y : R) (hx : x ∈ closure s) (hy : y ∈ closure s), p x y hx hy → p x (-y) hx ⋯
add_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : x ∈ closure s
z : R
hz : z ∈ s
⊢ p x z hx ⋯

Tactic:
induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | neg _ _ h => exact neg_left _ _ _ _ h

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

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
s : Set R
C : R → Prop
h1 : C 1
hneg1 : C (-1)
hs : ∀ z ∈ s, ∀ (n : R), C n → C (z * n)
ha : ∀ {x y : R}, C x → C y → C (x + y)
h0 : C 0
hd : R
tl : List R
ih : (∀ y ∈ tl, y ∈ s ∨ y = -1) → ∃ L, (∀ x ∈ L, x ∈ s) ∧ (tl.prod = L.prod ∨ tl.prod = -L.prod)
HL : (hd ∈ s ∨ hd = -1) ∧ ∀ x ∈ tl, x ∈ s ∨ x = -1
L : List R
HL' : ∀ x ∈ L, x ∈ s
HP : tl.prod = L.prod
hhd : hd ∈ s
⊢ (hd :: tl).prod = (hd :: L).prod

Tactic:
rw [List.prod_cons, List.prod_cons, HP]

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

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
a b : M
⊢ (∀ (a_1 : M),
      Associates.mk a_1 ∣ Associates.mk a → Associates.mk a_1 ∣ Associates.mk b → IsUnit (Associates.mk a_1)) ↔
    ∀ ⦃d : M⦄, d ∣ a → d ∣ b → IsUnit d

Tactic:
mk_dvd_mk,

Goal state after:
M : Type u_1
inst✝ : CommMonoid M
a b : M
⊢ (∀ (a_1 : M), a_1 ∣ a → a_1 ∣ b → IsUnit (Associates.mk a_1)) ↔ ∀ ⦃d : M⦄, d ∣ a → d ∣ b → IsUnit d
Module: Mathlib.Algebra.GroupWithZero.Associated

Goal state before:
pp: case succ.inl
M : Type u_1
inst✝¹ : CommMonoidWithZero M
inst✝ : IsCancelMulZero M
p : M
hp : Prime p
n : ℕ
ih : ∀ {q : M}, q ∣ p ^ n ↔ ∃ i, i ≤ n ∧ q ~ᵤ p ^ i
q : M
h : p * q ∣ p * p ^ n
⊢ ∃ i, i ≤ n + 1 ∧ p * q ~ᵤ p ^ i

Tactic:
rw [mul_dvd_mul_iff_left hp.ne_zero, ih] at h
      rcases h with ⟨i, hi, hq⟩
      refine ⟨i + 1, Nat.succ_le_succ hi, (hq.mul_left p).trans ?_⟩
      rw [pow_succ']

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set

Goal state before:
pp: case inl
α : Type u_1
inst✝ : GroupWithZero α
s : Set α
⊢ (0 •> s)⁻¹ = s⁻¹ <• 0⁻¹

Tactic:
obtain rfl | hs := s.eq_empty_or_nonempty

Goal state after:
case inl.inl
α : Type u_1
inst✝ : GroupWithZero α
⊢ (0 •> ∅)⁻¹ = ∅⁻¹ <• 0⁻¹

case inl.inr
α : Type u_1
inst✝ : GroupWithZero α
s : Set α
hs : s.Nonempty
⊢ (0 •> s)⁻¹ = s⁻¹ <• 0⁻¹
Module: Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set

Goal state before:
pp: case inl
α : Type u_1
inst✝ : GroupWithZero α
s : Set α
⊢ (s <• 0)⁻¹ = 0⁻¹ •> s⁻¹

Tactic:
obtain rfl | hs := s.eq_empty_or_nonempty

Goal state after:
case inl.inl
α : Type u_1
inst✝ : GroupWithZero α
⊢ (∅ <• 0)⁻¹ = 0⁻¹ •> ∅⁻¹

case inl.inr
α : Type u_1
inst✝ : GroupWithZero α
s : Set α
hs : s.Nonempty
⊢ (s <• 0)⁻¹ = 0⁻¹ •> s⁻¹
Module: Mathlib.Algebra.Module.Submodule.Range

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p q : Submodule R M
⊢ p.submoduleOf q = ⊤ ↔ q ≤ p

Tactic:
simp [submoduleOf]

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

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁵ : Semiring R
inst✝⁴ : Semiring R₂
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
f : M →ₛₗ[τ₁₂] M₂
h : ∀ (u v : ↥f.ker →ₗ[R] M), f ∘ₛₗ u = f ∘ₛₗ v → u = v
h₁ : f ∘ₛₗ 0 = 0
⊢ range 0 = ⊥

Tactic:
exact range_zero

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

Goal state before:
pp: case mp
R : Type u_2
M : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
S : Type u_4
inst✝¹ : Monoid S
inst✝ : DistribMulAction S M
N : Submodule R M
x✝ : M
⊢ x✝ ∈ ∅ • N → x✝ ∈ ⊥

Tactic:
intro hx
    rw [mem_set_smul_def, Submodule.mem_sInf] at hx
    exact hx ⊥ (fun r _ hr ↦ hr.elim)

Goal state after:
Module: Mathlib.Order.ModularLattice

Goal state before:
pp: α : Type u_1
inst✝¹ : Lattice α
inst✝ : IsModularLattice α
x✝ y✝ z a b : α
x y : ↑(Icc (a ⊓ b) a)
h : ↑x ⊔ b ≤ ↑y ⊔ b
⊢ ↑x ≤ ↑y

Tactic:
rw [← sup_eq_right.2 x.prop.1, inf_sup_assoc_of_le _ x.prop.2, sup_comm, ←
      sup_eq_right.2 y.prop.1, inf_sup_assoc_of_le _ y.prop.2, sup_comm b]

Goal state after:
α : Type u_1
inst✝¹ : Lattice α
inst✝ : IsModularLattice α
x✝ y✝ z a b : α
x y : ↑(Icc (a ⊓ b) a)
h : ↑x ⊔ b ≤ ↑y ⊔ b
⊢ a ⊓ (↑x ⊔ b) ≤ a ⊓ (↑y ⊔ b)
Module: Mathlib.Order.ModularLattice

Goal state before:
pp: case codisjoint
α : Type u_1
inst✝² : Lattice α
inst✝¹ : BoundedOrder α
inst✝ : IsModularLattice α
a b c : α
h₁ : IsCompl b c
h₂ : b ≤ a
⊢ Codisjoint ⟨a ⊓ b, ⋯⟩ ⟨a ⊓ c, ⋯⟩

Tactic:
· simp only [Iic.codisjoint_iff, inf_comm a, IsModularLattice.inf_sup_inf_assoc]
    simp [inf_of_le_left h₂, h₁.sup_eq_top]

Goal state after:
Module: Mathlib.Order.ModularLattice

Goal state before:
pp: α : Type u_1
inst✝³ : Lattice α
inst✝² : IsModularLattice α
a b c : α
inst✝¹ : BoundedOrder α
inst✝ : ComplementedLattice α
hb : a ≤ b
hc : b ≤ c
d : α
hdisjoint : Disjoint b d
hcodisjoint : Codisjoint b d
⊢ ∃ b', b ⊓ b' = a ∧ b ⊔ b' = c

Tactic:
refine ⟨(d ⊔ a) ⊓ c, ?_, ?_⟩

Goal state after:
case refine_1
α : Type u_1
inst✝³ : Lattice α
inst✝² : IsModularLattice α
a b c : α
inst✝¹ : BoundedOrder α
inst✝ : ComplementedLattice α
hb : a ≤ b
hc : b ≤ c
d : α
hdisjoint : Disjoint b d
hcodisjoint : Codisjoint b d
⊢ b ⊓ ((d ⊔ a) ⊓ c) = a

case refine_2
α : Type u_1
inst✝³ : Lattice α
inst✝² : IsModularLattice α
a b c : α
inst✝¹ : BoundedOrder α
inst✝ : ComplementedLattice α
hb : a ≤ b
hc : b ≤ c
d : α
hdisjoint : Disjoint b d
hcodisjoint : Codisjoint b d
⊢ b ⊔ (d ⊔ a) ⊓ c = c
Module: Mathlib.LinearAlgebra.Span.Defs

Goal state before:
pp: R : Type u_1
M : Type u_4
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Sort u_9
inst✝ : Nonempty ι
S : ι → Submodule R M
H : Directed (fun x1 x2 ↦ x1 ≤ x2) S
x : M
⊢ x ∈ ↑(iSup S) ↔ ∃ i, x ∈ S i

Tactic:
coe_iSup_of_directed S H,

Goal state after:
R : Type u_1
M : Type u_4
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Sort u_9
inst✝ : Nonempty ι
S : ι → Submodule R M
H : Directed (fun x1 x2 ↦ x1 ≤ x2) S
x : M
⊢ x ∈ ⋃ i, ↑(S i) ↔ ∃ i, x ∈ S i
Module: Mathlib.Order.Atoms

Goal state before:
pp: α : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : OrderBot α
a b : α
ha : IsAtom a
⊢ ¬Disjoint a b ↔ a ≤ b

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : SemilatticeInf α
inst✝ : OrderBot α
a b : α
ha : IsAtom a
⊢ ¬a ⊓ b = ⊥ ↔ a ≤ b
Module: Mathlib.LinearAlgebra.Span.Defs

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
x y : M
h : x ∈ R ∙ y
⊢ ∀ x ∈ {y}, ∃ a, a • y = x

Tactic:
rintro y (rfl | ⟨⟨_⟩⟩)
      exact ⟨1, by simp⟩

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

Goal state before:
pp: case neg
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
S : Set (Submodule R M) := {T | ∃ m ∈ p, m ≠ 0 ∧ T = R ∙ m}
m : M
hm : m ∈ p
h : ¬m = 0
⊢ m ∈ sSup {T | ∃ m ∈ p, m ≠ 0 ∧ T = R ∙ m}

Tactic:
exact @le_sSup _ _ S _ ⟨m, ⟨hm, ⟨h, rfl⟩⟩⟩ m (mem_span_singleton_self m)

Goal state after:
Module: Mathlib.Order.Atoms

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : OrderBot α
inst✝ : OrderBot β
f : β ↪o α
hbot : f ⊥ = ⊥
b : β
hb : IsAtom (f b)
⊢ IsAtom b

Tactic:
by
  simp only [← bot_covBy_iff] at hb ⊢
  exact CovBy.of_image f (hbot.symm ▸ hb)

Goal state after:
Module: Mathlib.Order.Atoms

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : OrderBot α
inst✝ : OrderBot β
f : α ≃o β
⊢ IsAtomic α ↔ IsAtomic β

Tactic:
simp only [isAtomic_iff, f.surjective.forall, f.surjective.exists, ← map_bot f, f.eq_iff_eq,
    f.le_iff_le, f.isAtom_iff]

Goal state after:
Module: Mathlib.Order.SupIndep

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝ : CompleteLattice α
t : ι → α
h : iSupIndep fun i ↦ t ↑i
⊢ iSupIndep t

Tactic:
simp only [iSupIndep_def] at h ⊢

Goal state after:
α : Type u_1
ι : Type u_3
inst✝ : CompleteLattice α
t : ι → α
h : ∀ (i : { i // t i ≠ ⊥ }), Disjoint (t ↑i) (⨆ j, ⨆ (_ : j ≠ i), t ↑j)
⊢ ∀ (i : ι), Disjoint (t i) (⨆ j, ⨆ (_ : j ≠ i), t j)
Module: Mathlib.Algebra.Module.Submodule.EqLocus

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁷ : Semiring R
inst✝⁶ : Semiring R₂
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M
inst✝² : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_5
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F τ₁₂ M M₂
f g : F
S T : Submodule R M
hS : Set.EqOn ⇑f ⇑g ↑S
hT : Set.EqOn ⇑f ⇑g ↑T
⊢ Set.EqOn ⇑f ⇑g ↑(S ⊔ T)

Tactic:
rw [← LinearMap.coe_coe (f := f), ← LinearMap.coe_coe (f := g), ← le_eqLocus] at hS hT ⊢
  exact sup_le hS hT

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

Goal state before:
pp: case refine_1
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : WellFoundedGT α
s : Set α
S : Set α := {x | ∃ t, ↑t ⊆ s ∧ t.sup id = x}
t : Finset α
ht₁ : ↑t ⊆ s
hm : ∀ x ∈ S, ¬t.sup id < x
y : α
hy : y ∈ s
⊢ y ≤ t.sup id

pp: case refine_2
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : WellFoundedGT α
s : Set α
S : Set α := {x | ∃ t, ↑t ⊆ s ∧ t.sup id = x}
t : Finset α
ht₁ : ↑t ⊆ s
hm : ∀ x ∈ S, ¬t.sup id < x
⊢ t.sup id ≤ sSup s

Tactic:
· classical
    rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
        (hm _ ⟨insert y t, by simp [Set.insert_subset_iff, hy, ht₁]⟩)]
    simp

Goal state after:
case refine_2
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : WellFoundedGT α
s : Set α
S : Set α := {x | ∃ t, ↑t ⊆ s ∧ t.sup id = x}
t : Finset α
ht₁ : ↑t ⊆ s
hm : ∀ x ∈ S, ¬t.sup id < x
⊢ t.sup id ≤ sSup s
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a : α
s : Set α
h : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ Disjoint a (sSup s) ↔ ∀ ⦃b : α⦄, b ∈ s → Disjoint a b

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a : α
s : Set α
h : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ a ⊓ sSup s = ⊥ ↔ ∀ ⦃b : α⦄, b ∈ s → a ⊓ b = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a : α
s : Set α
h : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ Disjoint (sSup s) a ↔ ∀ ⦃b : α⦄, b ∈ s → Disjoint b a

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a : α
s : Set α
h : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s
⊢ sSup s ⊓ a = ⊥ ↔ ∀ ⦃b : α⦄, b ∈ s → b ⊓ a = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: ι : Sort u_1
α : Type u_2
inst✝¹ : CompleteLattice α
f : ι → α
inst✝ : IsCompactlyGenerated α
a : α
h : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ Disjoint a (⨆ i, f i) ↔ ∀ (i : ι), Disjoint a (f i)

Tactic:
disjoint_iff,

Goal state after:
ι : Sort u_1
α : Type u_2
inst✝¹ : CompleteLattice α
f : ι → α
inst✝ : IsCompactlyGenerated α
a : α
h : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ a ⊓ ⨆ i, f i = ⊥ ↔ ∀ (i : ι), a ⊓ f i = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: ι : Sort u_1
α : Type u_2
inst✝¹ : CompleteLattice α
f : ι → α
inst✝ : IsCompactlyGenerated α
a : α
h : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ Disjoint (⨆ i, f i) a ↔ ∀ (i : ι), Disjoint (f i) a

Tactic:
disjoint_iff,

Goal state after:
ι : Sort u_1
α : Type u_2
inst✝¹ : CompleteLattice α
f : ι → α
inst✝ : IsCompactlyGenerated α
a : α
h : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ (⨆ i, f i) ⊓ a = ⊥ ↔ ∀ (i : ι), f i ⊓ a = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
s : Set α
h : ∀ (t : Finset α), ↑t ⊆ s → sSupIndep ↑t
a : α
ha : a ∈ s
⊢ Disjoint a (sSup (s \ {a}))

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
s : Set α
h : ∀ (t : Finset α), ↑t ⊆ s → sSupIndep ↑t
a : α
ha : a ∈ s
⊢ a ⊓ sSup (s \ {a}) = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
i x₁ : ι
hx₁ : ¬f x₁ = ⊥
x₂ : ι
h : Disjoint (f x₁) (f x₂)
hx₂ : ¬f x₂ = ⊥
hfeq : f x₁ = f x₂
hneq : ¬x₁ = x₂
⊢ False

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
i x₁ : ι
hx₁ : ¬f x₁ = ⊥
x₂ : ι
h : f x₁ ⊓ f x₂ = ⊥
hx₂ : ¬f x₂ = ⊥
hfeq : f x₁ = f x₂
hneq : ¬x₁ = x₂
⊢ False
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
h : ∀ (s : Finset ι), s.SupIndep f
i : ι
hf : InjOn f {i | f i ≠ ⊥}
⊢ Disjoint (f i) (sSup (f '' {j | j ≠ i}))

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
h : ∀ (s : Finset ι), s.SupIndep f
i : ι
hf : InjOn f {i | f i ≠ ⊥}
⊢ f i ⊓ sSup (f '' {j | j ≠ i}) = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
hf : InjOn f {i | f i ≠ ⊥}
h : ∀ (s : Finset ι), s.SupIndep f
i : ι
⊢ Disjoint (f i) (sSup (f '' {j | j ≠ i}))

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
hf : InjOn f {i | f i ≠ ⊥}
h : ∀ (s : Finset ι), s.SupIndep f
i : ι
⊢ f i ⊓ sSup (f '' {j | j ≠ i}) = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
s : Set ι
a : α
hs : ∀ t ⊆ s, t.Finite → Disjoint (⨆ i ∈ t, f i) a
⊢ Disjoint (⨆ i ∈ s, f i) a

Tactic:
disjoint_iff,

Goal state after:
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
ι : Type u_3
f : ι → α
s : Set ι
a : α
hs : ∀ t ⊆ s, t.Finite → Disjoint (⨆ i ∈ t, f i) a
⊢ (⨆ i ∈ s, f i) ⊓ a = ⊥
Module: Mathlib.Order.CompactlyGenerated.Basic

Goal state before:
pp: case inr.refine_1
α : Type u_2
inst✝ : CompleteLattice α
k : α
h : IsCompactElement k
b : α
hbk : b ∈ Iic k
H : b ≠ k
⊢ ∀ c ⊆ Iio k, IsChain (fun x1 x2 ↦ x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ Iio k, ∀ z ∈ c, z ≤ ub

Tactic:
intro S

Goal state after:
case inr.refine_1
α : Type u_2
inst✝ : CompleteLattice α
k : α
h : IsCompactElement k
b : α
hbk : b ∈ Iic k
H : b ≠ k
S : Set α
⊢ S ⊆ Iio k → IsChain (fun x1 x2 ↦ x1 ≤ x2) S → ∀ y ∈ S, ∃ ub ∈ Iio k, ∀ z ∈ S, z ≤ ub