mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₁ • b₁ = a₂ • b₂

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝¹ : Preorder α
inst✝ : Nonempty α
⊢ 1 < cof α ↔ NoTopOrder α

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₁ • b₁ = a₂ • b₂

Tactic:
by rintro ⟨rfl, rfl⟩; rfl

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb✝ : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
h : a₁ • b₁ = a₂ • b₂
hb : b₁ < b₂
⊢ a₁ • b₁ < a₂ • b₂

Tactic:
exact (smul_lt_smul_of_pos_left hb h₁).trans_le (smul_le_smul_of_nonneg_right ha h₂.le)

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₁ : 0 < a₁
h₂ : 0 < b₂
⊢ a₁ • b₁ = a₂ • b₂ ↔ a₁ = a₂ ∧ b₁ = b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₂ : 0 < a₂
h₁ : 0 < b₁
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₁ • b₁ = a₂ • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₂ : 0 < a₂
h₁ : 0 < b₁
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₁ • b₁ = a₂ • b₂

Tactic:
by rintro ⟨rfl, rfl⟩; rfl

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
S : Type u_3
P : Type u_4
inst✝³ : Semiring S
inst✝² : AddCommMonoid P
inst✝¹ : Module S P
σ : R →+* S
inst✝ : RingHomSurjective σ
f : M →ₛₗ[σ] P
hf : Function.Injective ⇑f
N : Submodule R M
hfn : (map f N).FG
t : Finset P
ht : span S ↑t = map f N
⊢ map f (span R ↑(t.preimage ⇑f ⋯)) = map f N

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
⊢ ⊤.FG ↔ N.FG

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a₁ a₂ : α
b₁ b₂ : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : PosSMulStrictMono α β
inst✝ : SMulPosStrictMono α β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
h₂ : 0 < a₂
h₁ : 0 < b₁
⊢ a₁ • b₁ = a₂ • b₂ ↔ a₁ = a₂ ∧ b₁ = b₂

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
hf : StrictMono f
hf' : IsCofinal (range f)
s : Set β
hs : IsCofinal s
x : ↑s
⊢ ∃ y, ↑x ≤ f y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: case inr.inl
α : Type u_1
β : Type u_2
b : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : LinearOrder α
inst✝² : LinearOrder β
inst✝¹ : PosSMulMono α β
inst✝ : SMulPosMono α β
hab : 0 < 0 • b
⊢ 0 < 0 ∧ 0 < b ∨ 0 < 0 ∧ b < 0

Tactic:
rw [zero_smul] at hab

Goal state after:
case inr.inl
α : Type u_1
β : Type u_2
b : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : LinearOrder α
inst✝² : LinearOrder β
inst✝¹ : PosSMulMono α β
inst✝ : SMulPosMono α β
hab : 0 < 0
⊢ 0 < 0 ∧ 0 < b ∨ 0 < 0 ∧ b < 0
Module: Mathlib.RingTheory.Finiteness.Basic

Goal state before:
pp: R : Type u_4
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
motive : (N : Submodule R M) → N.FG → Prop
singleton : ∀ (x : M), motive (R ∙ x) ⋯
sup : ∀ (N₁ N₂ : Submodule R M) (hN₁ : N₁.FG) (hN₂ : N₂.FG), motive N₁ hN₁ → motive N₂ hN₂ → motive (N₁ ⊔ N₂) ⋯
N : Submodule R M
hN : N.FG
⊢ motive N hN

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : SMulWithZero α β
inst✝³ : LinearOrder α
inst✝² : LinearOrder β
inst✝¹ : PosSMulMono α β
inst✝ : SMulPosMono α β
hab : 0 < a • b
⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0

Tactic:
by

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

Goal state before:
pp: R : Type u_4
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
motive : (N : Submodule R M) → N.FG → Prop
bot : motive ⊥ ⋯
sup : ∀ (N : Submodule R M) (x : M) (hN : N.FG), motive N hN → motive (N ⊔ R ∙ x) ⋯
x : M
s : Finset M
hxs : x ∉ s
ih : motive (span R ↑s) ⋯
⊢ (span R ↑s).FG

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁶ : Monoid α
inst✝⁵ : Zero β
inst✝⁴ : MulAction α β
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : SMulPosMono α β
inst✝ : SMulPosReflectLE α β
hb : 0 < b
⊢ b ≤ a • b ↔ 1 • b ≤ a • b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
hf : StrictMono f
hf' : IsCofinal (range f)
⊢ lift.{v, u} (cof α) = lift.{u, v} (cof β)

Tactic:
by

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

Goal state before:
pp: R : Type u_4
M : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
motive : (N : Submodule R M) → N.FG → Prop
bot : motive ⊥ ⋯
sup : ∀ (N : Submodule R M) (x : M) (hN : N.FG), motive N hN → motive (N ⊔ R ∙ x) ⋯
N : Submodule R M
hN : N.FG
⊢ motive N hN

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α γ : Type u
inst✝¹ : LinearOrder α
inst✝ : LinearOrder γ
f : α → γ
hf : StrictMono f
hf' : IsCofinal (range f)
⊢ cof α = cof γ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁶ : Monoid α
inst✝⁵ : Zero β
inst✝⁴ : MulAction α β
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : SMulPosStrictMono α β
inst✝ : SMulPosReflectLT α β
hb : 0 < b
⊢ b < a • b ↔ 1 • b < a • b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝ : LinearOrder α
s : Set α
hs : IsCofinal s
⊢ IsCofinal (range Subtype.val)

Tactic:
by

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

Goal state before:
pp: R : Type u_4
A : Type u_5
M : Type u_6
inst✝⁶ : Semiring A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module A M
S : Submodule A M
inst✝³ : CommSemiring R
inst✝² : Algebra R A
inst✝¹ : Module R M
inst✝ : IsScalarTower R A M
hS : S.FG
h : Surjective ⇑(algebraMap R A)
⊢ (restrictScalars R S).FG

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁶ : Monoid α
inst✝⁵ : Zero β
inst✝⁴ : MulAction α β
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : SMulPosMono α β
inst✝ : SMulPosReflectLE α β
hb : 0 < b
⊢ a • b ≤ b ↔ a • b ≤ 1 • b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝ : LinearOrder α
h : cof α < ℵ₀
s : Set α
hs : IsCofinal s
hs' : #↑s = cof α
⊢ s.Finite

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁶ : Monoid α
inst✝⁵ : Zero β
inst✝⁴ : MulAction α β
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : SMulPosStrictMono α β
inst✝ : SMulPosReflectLT α β
hb : 0 < b
⊢ a • b < b ↔ a • b < 1 • b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝ : LinearOrder α
⊢ cof α < ℵ₀ ↔ cof α ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝ : LinearOrder α
⊢ ℵ₀ ≤ cof α ↔ 1 < cof α

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝² : LinearOrder α
inst✝¹ : Nonempty α
inst✝ : NoMaxOrder α
⊢ ℵ₀ ≤ cof α

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁵ : Monoid α
inst✝⁴ : Zero β
inst✝³ : MulAction α β
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : SMulPosMono α β
hb : 0 ≤ b
h : a ≤ 1
⊢ a • b ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
inst✝³ : LinearOrder α
inst✝² : NoMaxOrder α
inst✝¹ : Nonempty α
inst✝ : Countable α
⊢ ℵ₀ ≤ cof α

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: ⊢ cof ℕ = ℵ₀

Tactic:
by

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

Goal state before:
pp: A : Type u_5
M : Type u_6
inst✝⁶ : Semiring A
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module A M
S : Submodule A M
R : Type u_7
inst✝³ : Semiring R
inst✝² : Module R M
inst✝¹ : SMul R A
inst✝ : IsScalarTower R A M
hS : (restrictScalars R S).FG
⊢ S.FG

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁵ : Monoid α
inst✝⁴ : Zero β
inst✝³ : MulAction α β
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : SMulPosMono α β
hb : 0 ≤ b
h : 1 ≤ a
⊢ b ≤ a • b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁵ : Monoid α
inst✝⁴ : Zero β
inst✝³ : MulAction α β
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : SMulPosStrictMono α β
hb : 0 < b
h : a < 1
⊢ a • b < b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : Preorder α
inst✝ : Preorder β
f : β → α
g : α → β
h : GaloisConnection f g
⊢ lift.{u, v} (cof β) ≤ lift.{v, u} (cof α)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
M' : Submodule R M
N : ℕ →o Submodule R M
H : iSup ⇑N = M'
S : Finset M
hS : span R ↑S = M'
s : ↥S
⊢ ↑s ∈ ⨆ k, N k

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α γ : Type u
inst✝¹ : Preorder α
inst✝ : Preorder γ
f : γ → α
g : α → γ
h : GaloisConnection f g
⊢ cof γ ≤ cof α

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α γ : Type u
inst✝¹ : Preorder α
inst✝ : Preorder γ
f : α ≃o γ
⊢ cof α = cof γ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b : β
inst✝⁵ : Monoid α
inst✝⁴ : Zero β
inst✝³ : MulAction α β
inst✝² : Preorder α
inst✝¹ : Preorder β
inst✝ : SMulPosStrictMono α β
hb : 0 < b
h : 1 < a
⊢ b < a • b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Semiring α
inst✝⁴ : AddCommGroup β
inst✝³ : Module α β
inst✝² : PartialOrder α
inst✝¹ : PartialOrder β
inst✝ : IsOrderedAddMonoid β
h : ∀ (a : α), 0 ≤ a → ∀ (b : β), 0 ≤ b → 0 ≤ a • b
_a : α
ha : 0 ≤ _a
b₁ b₂ : β
⊢ b₁ ≤ b₂ → _a • b₁ ≤ _a • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : Ring α
inst✝⁵ : AddCommGroup β
inst✝⁴ : Module α β
inst✝³ : PartialOrder α
inst✝² : PartialOrder β
inst✝¹ : IsOrderedAddMonoid α
inst✝ : IsOrderedAddMonoid β
h : ∀ (a : α), 0 ≤ a → ∀ (b : β), 0 ≤ b → 0 ≤ a • b
_b : β
hb : 0 ≤ _b
a₁ a₂ : α
⊢ a₁ ≤ a₂ → a₁ • _b ≤ a₂ • _b

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
s : Set (Set α)
h₁ : IsCofinal (⋃₀ s)
h₂ : #↑s < cof α
⊢ ∃ x ∈ s, IsCofinal x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b₁ b₂ : β
inst✝⁵ : GroupWithZero α
inst✝⁴ : Preorder α
inst✝³ : Preorder β
inst✝² : MulAction α β
inst✝¹ : PosSMulMono α β
inst✝ : PosSMulReflectLE α β
ha : 0 < a
⊢ a⁻¹ • b₁ ≤ b₂ ↔ b₁ ≤ a • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b₁ b₂ : β
inst✝⁵ : GroupWithZero α
inst✝⁴ : Preorder α
inst✝³ : Preorder β
inst✝² : MulAction α β
inst✝¹ : PosSMulMono α β
inst✝ : PosSMulReflectLE α β
ha : 0 < a
⊢ b₁ ≤ a⁻¹ • b₂ ↔ a • b₁ ≤ b₂

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
M' : Submodule R M
hM' : M'.FG
N : ℕ →o Submodule R M
H : iSup ⇑N = M'
⊢ ∃ n, M' = N n

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
sp : M → Submodule R M := fun a ↦ R ∙ a
t : Finset M
⊢ ⨆ x ∈ t, sp x = ⨆ x ∈ ↑t, sp x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b₁ b₂ : β
inst✝⁵ : GroupWithZero α
inst✝⁴ : Preorder α
inst✝³ : Preorder β
inst✝² : MulAction α β
inst✝¹ : PosSMulStrictMono α β
inst✝ : PosSMulReflectLT α β
ha : 0 < a
⊢ a⁻¹ • b₁ < b₂ ↔ b₁ < a • b₂

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Basic

Goal state before:
pp: α ι : Type u_1
inst✝ : LinearOrder α
s : ι → Set α
h₁ : IsCofinal (⋃ i, s i)
h₂ : #ι < cof α
⊢ ∃ i, IsCofinal (s i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b₁ b₂ : β
inst✝⁵ : GroupWithZero α
inst✝⁴ : Preorder α
inst✝³ : Preorder β
inst✝² : MulAction α β
inst✝¹ : PosSMulStrictMono α β
inst✝ : PosSMulReflectLT α β
ha : 0 < a
⊢ b₁ < a⁻¹ • b₂ ↔ a • b₁ < b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b b₁ b₂ : β
inst✝⁶ : Preorder α
inst✝⁵ : Monoid α
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : DistribMulAction α β
inst✝ : SMulPosMono α β
_b : βᵒᵈ
hb : 0 ≤ _b
a₁ a₂ : α
ha : a₁ ≤ a₂
⊢ a₁ • _b ≤ a₂ • _b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b b₁ b₂ : β
inst✝⁶ : Preorder α
inst✝⁵ : Monoid α
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : DistribMulAction α β
inst✝ : SMulPosStrictMono α β
_b : βᵒᵈ
hb : 0 < _b
a₁ a₂ : α
ha : a₁ < a₂
⊢ a₁ • _b < a₂ • _b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b b₁ b₂ : β
inst✝⁶ : Preorder α
inst✝⁵ : Monoid α
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : DistribMulAction α β
inst✝ : SMulPosReflectLT α β
_b : βᵒᵈ
hb : 0 ≤ _b
a₁ a₂ : α
h : a₁ • _b < a₂ • _b
⊢ a₁ < a₂

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
sp : M → Submodule R M := fun a ↦ R ∙ a
supr_rw : ∀ (t : Finset M), ⨆ x ∈ t, sp x = ⨆ x ∈ ↑t, sp x
h : ∀ (s_1 : Set (Submodule R M)), s ≤ sSup s_1 → ∃ t, ↑t ⊆ s_1 ∧ s ≤ t.sup id
⊢ s = sSup (sp '' ↑s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b b₁ b₂ : β
inst✝⁶ : Preorder α
inst✝⁵ : Monoid α
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : DistribMulAction α β
inst✝ : SMulPosReflectLE α β
_b : βᵒᵈ
hb : 0 < _b
a₁ a₂ : α
h : a₁ • _b ≤ a₂ • _b
⊢ a₁ ≤ a₂

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
sp : M → Submodule R M := fun a ↦ R ∙ a
supr_rw : ∀ (t : Finset M), ⨆ x ∈ t, sp x = ⨆ x ∈ ↑t, sp x
h : ∀ (s_1 : Set (Submodule R M)), s ≤ sSup s_1 → ∃ t, ↑t ⊆ s_1 ∧ s ≤ t.sup id
sSup' : s = sSup (sp '' ↑s)
u : Finset (Submodule R M)
huspan : ↑u ⊆ sp '' ↑s
husup : s ≤ u.sup id
⊢ s = u.sup id

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁸ : Semiring α
inst✝⁷ : PartialOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : PartialOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
a₁ a₂ : α
b₁ b₂ : β
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
⊢ a₁ • b₂ + a₂ • b₁ ≤ a₁ • b₁ + a₂ • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁸ : Semiring α
inst✝⁷ : PartialOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : PartialOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
a₁ a₂ : α
b₁ b₂ : β
ha : a₂ ≤ a₁
hb : b₂ ≤ b₁
⊢ a₁ • b₂ + a₂ • b₁ ≤ a₁ • b₁ + a₂ • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : Preorder α
⊢ DirSupClosedOn univ s ↔ DirSupClosed s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : Preorder α
⊢ DirSupInaccOn univ s ↔ DirSupInacc s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝¹ : Preorder α
inst✝ : IsEmpty α
s : Set α
⊢ DirSupClosedOn D s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝¹ : Preorder α
inst✝ : IsEmpty α
s : Set α
⊢ DirSupInaccOn D s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁸ : Semiring α
inst✝⁷ : PartialOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : PartialOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulStrictMono α β
a₁ a₂ : α
b₁ b₂ : β
ha : a₁ < a₂
hb : b₁ < b₂
⊢ a₁ • b₂ + a₂ • b₁ < a₁ • b₁ + a₂ • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
s : Set α
D : Set (Set α)
inst✝ : Preorder α
⊢ DirSupClosedOn D sᶜ ↔ DirSupInaccOn D s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁸ : Semiring α
inst✝⁷ : PartialOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : PartialOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulStrictMono α β
a₁ a₂ : α
b₁ b₂ : β
ha : a₂ < a₁
hb : b₂ < b₁
⊢ a₁ • b₂ + a₂ • b₁ < a₁ • b₁ + a₂ • b₂

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : Preorder α
⊢ DirSupClosed sᶜ ↔ DirSupInacc s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
s : Set α
D : Set (Set α)
inst✝ : Preorder α
⊢ DirSupInaccOn D sᶜ ↔ DirSupClosedOn D s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
s : Set α
inst✝ : Preorder α
⊢ DirSupInacc sᶜ ↔ DirSupClosed s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
⊢ DirSupClosed ∅

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
⊢ DirSupInacc ∅

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝ : Preorder α
⊢ DirSupClosedOn D ∅

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝ : Preorder α
⊢ DirSupInaccOn D ∅

Tactic:
by

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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
⊢ s.FG ↔ IsCompactElement s

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
⊢ DirSupClosed Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
⊢ DirSupInacc Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝ : Preorder α
⊢ DirSupClosedOn D Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝ : Preorder α
⊢ DirSupInaccOn D Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a : α
b₁ b₂ : β
inst✝⁷ : Ring α
inst✝⁶ : PartialOrder α
inst✝⁵ : IsOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
h : b₁ ≤ b₂
ha : a ≤ 0
⊢ a • b₂ ≤ a • b₁

Tactic:
by

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

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
⊢ IsCompactElement ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : Set (Set α)
hs : ∀ x ∈ s, DirSupClosed x
⊢ DirSupClosed (⋂₀ s)

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
D : Set (Set α)
inst✝ : Preorder α
s : Set (Set α)
hs : ∀ x ∈ s, DirSupInaccOn D x
⊢ DirSupInaccOn D (⋃₀ s)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Module.Defs

Goal state before:
pp: α : Type u_1
β : Type u_2
a a₁✝ a₂✝ : α
b b₁ b₂ : β
inst✝⁷ : Ring α
inst✝⁶ : PartialOrder α
inst✝⁵ : IsOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : PartialOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
_b : β
hb : 0 ≤ _b
a₁ a₂ : α
ha : a₁ ≤ a₂
⊢ a₁ • _b ≤ a₂ • _b

Tactic:
by

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

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : Finite M
val✝ : Fintype M
⊢ span R ↑Finset.univ = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Order.DirSupClosed

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
s : Set (Set α)
hs : ∀ x ∈ s, DirSupInacc x
⊢ DirSupInacc (⋃₀ s)

Tactic:
by

Goal state after: