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

Goal state before:
pp: case inl
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
this : ∀ (x : α), 0 < x → (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
x : α
h : x < 0
⊢ (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x

pp: case inr.inl
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
this : ∀ (x : α), 0 < x → (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
⊢ (inducedAddHom α β) (0 * 0) = (inducedAddHom α β) 0 * (inducedAddHom α β) 0

pp: case inr.inr
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
this : ∀ (x : α), 0 < x → (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
x : α
h : 0 < x
⊢ (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x

Tactic:
· convert! this (-x) (neg_pos.2 h) using 1
          · rw [neg_mul, mul_neg, neg_neg]
          · simp_rw [map_neg, neg_mul, mul_neg, neg_neg]

Goal state after:
case inr.inl
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
this : ∀ (x : α), 0 < x → (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
⊢ (inducedAddHom α β) (0 * 0) = (inducedAddHom α β) 0 * (inducedAddHom α β) 0

case inr.inr
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁹ : Field α
inst✝⁸ : LinearOrder α
inst✝⁷ : IsStrictOrderedRing α
inst✝⁶ : Field β
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : IsStrictOrderedRing β
inst✝³ : Field γ
inst✝² : ConditionallyCompleteLinearOrder γ
inst✝¹ : IsStrictOrderedRing γ
inst✝ : Archimedean α
a : α
b : β
q : ℚ
this : ∀ (x : α), 0 < x → (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
x : α
h : 0 < x
⊢ (inducedAddHom α β) (x * x) = (inducedAddHom α β) x * (inducedAddHom α β) x
Module: Mathlib.Algebra.Order.Chebyshev

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semifield α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f : ι → α
hf : ∀ i ∈ s, 0 ≤ f i
n : ℕ
hs : s.Nonempty
⊢ 0 < ↑(#s) ^ n

Tactic:
positivity

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semifield α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f : ι → α
hs : s.Nonempty
⊢ 0 < ↑(#s) ^ 2

Tactic:
positivity

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Semifield α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
s : Finset ι
f : ι → α
hs : s.Nonempty
⊢ 0 < ↑(#s)

Tactic:
positivity

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

Goal state before:
pp: case pos
α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
i : ι
hi : IsMax i
⊢ (partialSups f) (i + 1) = (partialSups f) i ⊔ disjointed f (i + 1)

Tactic:
have : i + 1 = i := by
      have h : i ≤ i + 1 := by
        rw [← Order.succ_eq_add_one]
        apply Order.le_succ
      exact le_antisymm (hi h) h

Goal state after:
case pos
α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
i : ι
hi : IsMax i
this : i + 1 = i
⊢ (partialSups f) (i + 1) = (partialSups f) i ⊔ disjointed f (i + 1)
Module: Mathlib.NumberTheory.ArithmeticFunction.Defs

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ f (x.lcm y) * f (x.gcd y) = f x * f y

Tactic:
iterate 4 rw [hf.multiplicative_factorization f (by assumption),
    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)
      (s := (x.primeFactors ∪ y.primeFactors))]

Goal state after:
case neg
R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors, f (x_1 ^ (x.lcm y).factorization x_1)) *
      ∏ x_1 ∈ x.primeFactors ∪ y.primeFactors, f (x_1 ^ (x.gcd y).factorization x_1) =
    (∏ x_1 ∈ x.primeFactors ∪ y.primeFactors, f (x_1 ^ x.factorization x_1)) *
      ∏ x ∈ x.primeFactors ∪ y.primeFactors, f (x ^ y.factorization x)

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ x.factorization.support ⊆ x.primeFactors ∪ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.gcd y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors

R : Type u_1
inst✝ : CommMonoidWithZero R
f : ArithmeticFunction R
hf : f.IsMultiplicative
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hgcd_ne_zero : x.gcd y ≠ 0
hlcm_ne_zero : x.lcm y ≠ 0
hfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1
⊢ (x.lcm y).factorization.support ⊆ x.primeFactors ∪ y.primeFactors
Module: Mathlib.Algebra.Order.Disjointed

Goal state before:
pp: case zero
α : Type u_1
ι : Type u_2
inst✝ : GeneralizedBooleanAlgebra α
f : ℕ → α
p : α → Sort u_3
hdiff : ⦃t : α⦄ → ⦃i : ℕ⦄ → p t → p (t \ f i)
n : ℕ
h : p (f (n + 1))
⊢ p (f (n + 1) \ (partialSups f) 0)

Tactic:
exact hdiff h

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

Goal state before:
pp: case inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulStrictMono α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
x : ι
hx : x ∈ ↑s
y : ι
hy : y ∈ ↑s
hgxy : (g ∘ ⇑σ) x < (g ∘ ⇑σ) y
hfxy : f y < f x
τ : Perm ι := Equiv.trans (Equiv.swap x y) σ
hτs : {x | τ x ≠ x} ⊆ ↑s
hxy : x ≠ y
⊢ ∑ i ∈ (s.erase x).erase y, f i • g (σ i) + (f y • g (σ y) + f x • g (σ x)) <
    ∑ i ∈ (s.erase x).erase y, f i • g (σ ((Equiv.swap x y) i)) + (f y • g (σ x) + f x • g (σ y))

Tactic:
refine add_lt_add_of_le_of_lt (Finset.sum_congr rfl fun z hz ↦ ?_).le
    (smul_add_smul_lt_smul_add_smul hfxy hgxy)

Goal state after:
case inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁸ : Semiring α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : ExistsAddOfLE α
inst✝⁴ : AddCommMonoid β
inst✝³ : LinearOrder β
inst✝² : IsOrderedCancelAddMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulStrictMono α β
s : Finset ι
σ : Perm ι
f : ι → α
g : ι → β
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
x : ι
hx : x ∈ ↑s
y : ι
hy : y ∈ ↑s
hgxy : (g ∘ ⇑σ) x < (g ∘ ⇑σ) y
hfxy : f y < f x
τ : Perm ι := Equiv.trans (Equiv.swap x y) σ
hτs : {x | τ x ≠ x} ⊆ ↑s
hxy : x ≠ y
z : ι
hz : z ∈ (s.erase x).erase y
⊢ f z • g (σ z) = f z • g (σ ((Equiv.swap x y) z))
Module: Mathlib.Algebra.Order.Archimedean.Class

Goal state before:
pp: M : Type u_1
inst✝² : CommGroup M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedMonoid M
a b : M
h : mk a < mk b
hpos : 1 ≤ a
⊢ b < a

Tactic:
by
  obtain h := mk_lt_mk.mp h 1
  rw [pow_one, mabs_lt, mabs_eq_self.mpr hpos] at h
  exact h.2

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Semifield K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : FloorSemiring K
a : K
n : ℕ
hn : n > 0
⊢ ↑n ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hab : a ≤ (b - 1)⁻¹
⊢ 0 < b

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ 0 ≤ (b - 1)⁻¹

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
a b : K
hb : 0 < b - 1
hba✝ : ↑⌈(b - 1)⁻¹⌉₊ / b < a
hba : (b - 1)⁻¹ ≤ a
⊢ b - 1 ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊ / b
⊢ 0 ≤ ↑⌈(b - 1)⁻¹⌉₊

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Semifield

Goal state before:
pp: case inl.hb
K : Type u_2
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsOrderedRing K
inst✝ : FloorSemiring K
b : K
hb : 1 < b
hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ ↑⌈(b - 1)⁻¹⌉₊ / b
⊢ b ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0
n : ℕ
⊢ ⌊↑r - ↑n⌋ₑ = ↑(⌊r⌋₊ - n)

Tactic:
← ENNReal.ofNNReal_sub_natCast

Goal state after:
r : ℝ≥0
n : ℕ
⊢ ⌊↑(r - ↑n)⌋ₑ = ↑(⌊r⌋₊ - n)
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0
n : ℕ
⊢ ⌈↑r - ↑n⌉ₑ = ↑(⌈r⌉₊ - n)

Tactic:
← ENNReal.ofNNReal_sub_natCast

Goal state after:
r : ℝ≥0
n : ℕ
⊢ ⌈↑(r - ↑n)⌉ₑ = ↑(⌈r⌉₊ - n)
Module: Mathlib.Algebra.Order.Floor.Extended

Goal state before:
pp: r : ℝ≥0∞
hr : 0 < r
⊢ 0 < ⌈r⌉ₑ

Tactic:
positivity

Goal state after:
Module: Mathlib.Algebra.Order.Group.Cone

Goal state before:
pp: S : Type u_1
G : Type u_2
inst✝² : CommGroup G
inst✝¹ : SetLike S G
C : S
inst✝ : GroupConeClass S G
x✝ : PartialOrder G := PartialOrder.mkOfGroupCone C
a b : G
nab : a ≤ b
c : G
⊢ a * c ≤ b * c

Tactic:
simpa [· ≤ ·] using nab

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

Goal state before:
pp: case refine_1
M : Type u_1
inst✝⁸ : AddCommGroup M
inst✝⁷ : LinearOrder M
inst✝⁶ : IsOrderedAddMonoid M
K : Type u_2
inst✝⁵ : Ring K
inst✝⁴ : LinearOrder K
inst✝³ : IsOrderedRing K
inst✝² : Archimedean K
inst✝¹ : Module K M
inst✝ : PosSMulMono K M
a : M
k : K
h : k ≠ 0
this : Nontrivial K
m : ℕ
hm : 1 ≤ m • |k|
n : ℕ
hn : |k| ≤ n • 1
⊢ |a| ≤ m • |k| • |a|

Tactic:
rw [← smul_assoc]

Goal state after:
case refine_1
M : Type u_1
inst✝⁸ : AddCommGroup M
inst✝⁷ : LinearOrder M
inst✝⁶ : IsOrderedAddMonoid M
K : Type u_2
inst✝⁵ : Ring K
inst✝⁴ : LinearOrder K
inst✝³ : IsOrderedRing K
inst✝² : Archimedean K
inst✝¹ : Module K M
inst✝ : PosSMulMono K M
a : M
k : K
h : k ≠ 0
this : Nontrivial K
m : ℕ
hm : 1 ≤ m • |k|
n : ℕ
hn : |k| ≤ n • 1
⊢ |a| ≤ (m • |k|) • |a|
Module: Mathlib.Data.Set.MulAntidiagonal

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).1 = (↑y).1
⊢ a = (↑y).1 * (↑x).2

Tactic:
rw [← h]

Goal state after:
α : Type u_1
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).1 = (↑y).1
⊢ a = (↑x).1 * (↑x).2
Module: Mathlib.Data.Set.MulAntidiagonal

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).2 = (↑y).2
⊢ a = (↑x).1 * (↑y).2

Tactic:
rw [← h]

Goal state after:
α : Type u_1
inst✝¹ : CommMonoid α
inst✝ : IsCancelMul α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).2 = (↑y).2
⊢ a = (↑x).1 * (↑x).2
Module: Mathlib.Algebra.Order.Interval.Set.SuccPred

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : One α
inst✝¹ : Add α
inst✝ : SuccAddOrder α
a b : α
h : a < b
⊢ insert (a + 1) (Ioc (a + 1) b) = Ioc a b

Tactic:
by
  simpa [succ_eq_add_one] using insert_Ioc_succ_left_eq_Ioc h

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

Goal state before:
pp: case refine_1.refine_1
α : Type u_2
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
s t : NonemptyInterval α
h : (s * t).toProd.1 = (toProd 1).1 ∧ (s * t).toProd.2 = (toProd 1).2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ s.toProd = (pure s.toProd.1).toProd

Tactic:
dsimp [pure]

Goal state after:
case refine_1.refine_1
α : Type u_2
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
s t : NonemptyInterval α
h : (s * t).toProd.1 = (toProd 1).1 ∧ (s * t).toProd.2 = (toProd 1).2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ s.toProd = (s.toProd.1, s.toProd.1)
Module: Mathlib.Algebra.Order.Interval.Basic

Goal state before:
pp: case refine_1.refine_2
α : Type u_2
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
s t : NonemptyInterval α
h : (s * t).toProd.1 = (toProd 1).1 ∧ (s * t).toProd.2 = (toProd 1).2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ t.toProd = (pure t.toProd.1).toProd

Tactic:
dsimp [pure]

Goal state after:
case refine_1.refine_2
α : Type u_2
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
s t : NonemptyInterval α
h : (s * t).toProd.1 = (toProd 1).1 ∧ (s * t).toProd.2 = (toProd 1).2
this : s.toProd.1 = s.toProd.2 ∧ t.toProd.1 = t.toProd.2
⊢ t.toProd = (t.toProd.1, t.toProd.1)
Module: Mathlib.Algebra.Order.Interval.Basic

Goal state before:
pp: case refine_2
α : Type u_2
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
b c : α
h : b * c = 1
⊢ pure b * pure c = 1

Tactic:
pure_mul_pure,

Goal state after:
case refine_2
α : Type u_2
inst✝² : CommGroup α
inst✝¹ : PartialOrder α
inst✝ : IsOrderedMonoid α
b c : α
h : b * c = 1
⊢ pure (b * c) = 1
Module: Mathlib.Order.Interval.Basic

Goal state before:
pp: case neg.inl
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : CompleteLattice α
inst✝ : DecidableLE α
s₁ : Set (Interval α)
s : NonemptyInterval α
ha : ↑s ∈ lowerBounds s₁
h : ⊥ ∈ s₁
⊢ ↑s ≤ ⊥

Tactic:
exact ha h

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case inl
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hle : x ≤ 0
⊢ (ofLex |x|).support = (ofLex x).support

pp: case inr
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hge : 0 ≤ x
⊢ (ofLex |x|).support = (ofLex x).support

Tactic:
· rw [abs_eq_neg_self.mpr hle]
    simp

Goal state after:
case inr
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x : Lex R⟦Γ⟧
hge : 0 ≤ x
⊢ (ofLex |x|).support = (ofLex x).support
Module: Mathlib.Algebra.Order.Quantale

Goal state before:
pp: α : Type u_1
x y z : α
inst✝² : Semigroup α
inst✝¹ : CompleteLattice α
inst✝ : IsQuantale α
h1 : x ≤ y ⇨ᵣ z
⊢ y * x ≤ z

Tactic:
by
    grw [h1]
    simp_all only [rightMulResiduation, mul_sSup_distrib, Set.mem_ofPred_eq,
      iSup_le_iff, implies_true]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case mk
R : Type u_1
S : Type u_2
inst✝³ : LinearOrder R
inst✝² : LinearOrder S
inst✝¹ : CommRing R
inst✝ : IsStrictOrderedRing R
n : ℕ
x : R
⊢ (n + 1) • mk x = n • mk x + mk x

Tactic:
rw [← mk_pow, pow_succ, mk_mul, mk_pow]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case inr
α : Type u_2
S : Type u_3
inst✝¹ : LinearOrder S
inst✝ : AddCommGroup α
f : α → S
fna : IsNonarchimedean f
Neg : ∀ (a : α), f a = f (-a)
a b : α
hne : f a ≠ f b
hab : f a > f b
h : f (a + b + -b) ≤ f (-b)
⊢ max (f a) (f b) ≤ f (a + b)

Tactic:
· exact absurd h (not_le.mpr (by simpa [Neg b] using hab))

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

Goal state before:
pp: K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
seed : Seed K M R
x : ↥seed.baseEmbedding.domain
hx : 0 < x
hmem : ↑x ∈ seed.baseDomain
f : Π₀ (i : FiniteArchimedeanClass M), ↥(seed.stratum i)
hf : (f.sum fun x xi ↦ ↑xi) = ↑x
hfpos : 0 < f.sum fun x x_1 ↦ ↑x_1
hsupport : f.support.Nonempty
hferase :
  (f.sum fun x x_1 ↦ ↑x_1) - ↑(f (f.support.min' hsupport)) = ∑ x ∈ f.support.erase (f.support.min' hsupport), ↑(f x)
c : FiniteArchimedeanClass M
hc : c ∈ ↑f.support
d : FiniteArchimedeanClass M
hd : d ∈ ↑f.support
h : c < d
⊢ ArchimedeanClass.mk ↑(f c) < ArchimedeanClass.mk ↑(f d)

Tactic:
rw [seed.archimedeanClassMk_of_mem_stratum (f c).prop (by simpa using! hc)]

Goal state after:
K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
seed : Seed K M R
x : ↥seed.baseEmbedding.domain
hx : 0 < x
hmem : ↑x ∈ seed.baseDomain
f : Π₀ (i : FiniteArchimedeanClass M), ↥(seed.stratum i)
hf : (f.sum fun x xi ↦ ↑xi) = ↑x
hfpos : 0 < f.sum fun x x_1 ↦ ↑x_1
hsupport : f.support.Nonempty
hferase :
  (f.sum fun x x_1 ↦ ↑x_1) - ↑(f (f.support.min' hsupport)) = ∑ x ∈ f.support.erase (f.support.min' hsupport), ↑(f x)
c : FiniteArchimedeanClass M
hc : c ∈ ↑f.support
d : FiniteArchimedeanClass M
hd : d ∈ ↑f.support
h : c < d
⊢ ↑c < ArchimedeanClass.mk ↑(f d)
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : CommRing R
inst✝⁴ : IsStrictOrderedRing R
S : Type u_3
inst✝³ : LinearOrder S
inst✝² : CommRing S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
f : S →+o R
x : S
h : x ≠ 0
⊢ (orderHom f) (mk x) = mk (f 1)

Tactic:
mk_eq_zero_of_archimedean h,

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : CommRing R
inst✝⁴ : IsStrictOrderedRing R
S : Type u_3
inst✝³ : LinearOrder S
inst✝² : CommRing S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
f : S →+o R
x : S
h : x ≠ 0
⊢ (orderHom f) 0 = mk (f 1)
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : LinearOrder R
inst✝² : LinearOrder S
inst✝¹ : Field R
inst✝ : IsOrderedRing R
⊢ -⊤ = ⊤

Tactic:
simp [← mk_zero, ← mk_inv]

Goal state after:
Module: Mathlib.Algebra.Order.Star.Pi

Goal state before:
pp: case refine_1
ι : Type u_1
inst✝⁴ : Finite ι
A : ι → Type u_2
inst✝³ : (i : ι) → PartialOrder (A i)
inst✝² : (i : ι) → NonUnitalSemiring (A i)
inst✝¹ : (i : ι) → StarRing (A i)
inst✝ : ∀ (i : ι), StarOrderedRing (A i)
xa xy : (i : ι) → A i
this : closure (Set.range fun s ↦ star s * s) = pi Set.univ fun i ↦ closure (Set.range fun s ↦ star s * s)
h : ∀ (i : ι), ∃ p ∈ closure (Set.range fun s ↦ star s * s), xy i = xa i + p
⊢ ∃ p, (∀ (i : ι), p i ∈ closure (Set.range fun s ↦ star s * s)) ∧ ∀ (x : ι), xy x = xa x + p x

Tactic:
exact ⟨fun i => h i |>.choose, fun i => h i |>.choose_spec.1, fun i => h i |>.choose_spec.2⟩

Goal state after:
Module: Mathlib.Algebra.Order.Star.Pi

Goal state before:
pp: ι : Type u_1
inst✝⁴ : Finite ι
A : ι → Type u_2
inst✝³ : (i : ι) → PartialOrder (A i)
inst✝² : (i : ι) → NonUnitalSemiring (A i)
inst✝¹ : (i : ι) → StarRing (A i)
inst✝ : ∀ (i : ι), StarOrderedRing (A i)
xa xy : (i : ι) → A i
⊢ xa ≤ xy ↔ ∃ p ∈ closure (Set.range fun s ↦ star s * s), xy = xa + p

Tactic:
have : closure (Set.range fun s : Π i, A i ↦ star s * s) =
        pi Set.univ fun i => (closure <| Set.range fun s : A i ↦ star s * s) := by
      rw [← closure_pi fun _ => Set.mem_range.mpr ⟨0, by simp⟩]
      congr
      ext x
      simp only [Set.mem_range, funext_iff, mul_apply, star_apply, Set.mem_pi,
        Set.mem_univ, forall_const]
      exact ⟨fun ⟨y, hy⟩ i => ⟨y i, hy i⟩, fun h => ⟨fun i => h i |>.choose,
        fun i => h i |>.choose_spec⟩⟩
    simp only [this, Pi.le_def, StarOrderedRing.le_iff, mem_pi, Set.mem_univ, forall_const]
    refine ⟨fun h => ?_, ?_⟩
    · simp only [funext_iff, add_apply]
      exact ⟨fun i => h i |>.choose, fun i => h i |>.choose_spec.1, fun i => h i |>.choose_spec.2⟩
    · simp only [forall_exists_index, and_imp]
      intro x h rfl i
      exact ⟨x i, by simp [h]⟩

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationSubring

Goal state before:
pp: case neg.inl
K : Type u
inst✝ : Field K
A B : ValuationSubring K
h : A.unitGroup ≤ B.unitGroup
x : K
h_1 : ¬x = 0
h_2 : ¬1 + x = 0
hx : A.valuation x < 1
⊢ x ∈ B

pp: case neg.inr
K : Type u
inst✝ : Field K
A B : ValuationSubring K
h : A.unitGroup ≤ B.unitGroup
x : K
h_1 : ¬x = 0
h_2 : ¬1 + x = 0
hx : A.valuation x = 1
⊢ x ∈ B

Tactic:
· have := h (show Units.mk0 _ h_2 ∈ A.unitGroup from A.valuation.map_one_add_of_lt hx)
      simpa using
        B.add_mem _ _ (show 1 + x ∈ B from SetLike.coe_mem (B.unitGroupMulEquiv ⟨_, this⟩ : B))
          (B.neg_mem _ B.one_mem)

Goal state after:
case neg.inr
K : Type u
inst✝ : Field K
A B : ValuationSubring K
h : A.unitGroup ≤ B.unitGroup
x : K
h_1 : ¬x = 0
h_2 : ¬1 + x = 0
hx : A.valuation x = 1
⊢ x ∈ B