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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝³ : CommSemiring k
inst✝² : CommMonoid G
inst✝¹ : MulSemiringAction G k
inst✝ : SMulCommClass G k k
a b : SkewMonoidAlgebra k G
hgk : ∀ (g : G) (r : k), g • r = r
x : G
x✝¹ : x ∈ b.coeff.support
y : G
x✝ : y ∈ a.coeff.support
⊢ single (y * x) (a.coeff y * b.coeff x) = single (x * y) (b.coeff x * a.coeff y)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝³ : CommSemiring k
inst✝² : CommMonoid G
inst✝¹ : MulSemiringAction G k
inst✝ : SMulCommClass G k k
a b : SkewMonoidAlgebra k G
⊢ a * b = b * a

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
inst✝³ : AddMonoid k
inst✝² : SMulZeroClass S k
inst✝¹ : FaithfulSMul S k
inst✝ : Nonempty G
_s₁ _s₂ : S
h : ∀ (a : SkewMonoidAlgebra k G), _s₁ • a = _s₂ • a
⊢ _s₁ = _s₂

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
inst✝⁴ : AddMonoid k
inst✝³ : SMul S₁ S₂
inst✝² : SMulZeroClass S₁ k
inst✝¹ : SMulZeroClass S₂ k
inst✝ : IsScalarTower S₁ S₂ k
x✝² : S₁
x✝¹ : S₂
x✝ : SkewMonoidAlgebra k G
coeff✝ : G →₀ k
⊢ (x✝² • x✝¹) • { coeff := coeff✝ } = x✝² • x✝¹ • { coeff := coeff✝ }

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
inst✝³ : AddMonoid k
inst✝² : SMulZeroClass S₁ k
inst✝¹ : SMulZeroClass S₂ k
inst✝ : SMulCommClass S₁ S₂ k
x✝² : S₁
x✝¹ : S₂
x✝ : SkewMonoidAlgebra k G
coeff✝ : G →₀ k
⊢ x✝² • x✝¹ • { coeff := coeff✝ } = x✝¹ • x✝² • { coeff := coeff✝ }

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
inst✝³ : AddMonoid k
inst✝² : SMulZeroClass S k
inst✝¹ : SMulZeroClass Sᵐᵒᵖ k
inst✝ : IsCentralScalar S k
x✝¹ : S
x✝ : SkewMonoidAlgebra k G
coeff✝ : G →₀ k
⊢ MulOpposite.op x✝¹ • { coeff := coeff✝ } = x✝¹ • { coeff := coeff✝ }

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
inst✝² : Semiring S
inst✝¹ : AddCommMonoid k
inst✝ : Module S k
⊢ ∀ (c : S) (x : SkewMonoidAlgebra k G), coeffAddEquiv (c • x) = c • coeffAddEquiv x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.UnitaryStarAlgAut

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : StarMul R
inst✝⁶ : CommRing S
inst✝⁵ : StarMul S
inst✝⁴ : Algebra S R
inst✝³ : StarModule S R
inst✝² : Algebra.IsCentral S R
inst✝¹ : IsCancelMulZero S
inst✝ : Module.IsTorsionFree S R
u v : ↥(unitary R)
x✝ : ∃ y, y • 1 = star ↑v * ↑u
y : S
h : y • 1 = star ↑v * ↑u
huv : ↑u = y • ↑v
hvu : ↑v = star y • ↑u
this : star y * y = 1
⊢ y ∈ unitary S

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
M : Type u_6
α : Type u_7
inst✝² : Monoid G
inst✝¹ : AddCommMonoid M
inst✝ : MulAction G α
g g' : G
f : SkewMonoidAlgebra M α
⊢ (g * g') • f = g • g' • f

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
M : Type u_6
α : Type u_7
inst✝² : Monoid G
inst✝¹ : AddCommMonoid M
inst✝ : MulAction G α
f : SkewMonoidAlgebra M α
⊢ 1 • f = f

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.UnitaryStarAlgAut

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : StarMul R
inst✝⁶ : CommRing S
inst✝⁵ : StarMul S
inst✝⁴ : Algebra S R
inst✝³ : StarModule S R
inst✝² : Algebra.IsCentral S R
inst✝¹ : IsCancelMulZero S
inst✝ : Module.IsTorsionFree S R
u v : ↥(unitary R)
x✝ : ∃ y, y • 1 = star ↑v * ↑u
y : S
h : y • 1 = star ↑v * ↑u
huv : ↑u = y • ↑v
hvu : ↑v = star y • ↑u
this : star y * y = 1
⊢ u = ⟨y, ⋯⟩ • v

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
M : Type u_6
α : Type u_7
inst✝³ : Monoid G
inst✝² : AddCommMonoid M
inst✝¹ : MulAction G α
inst✝ : AddCommMonoid k
g : G
⊢ g • 0 = 0

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
S : Type u_3
S₁ : Type u_4
S₂ : Type u_5
M : Type u_6
α : Type u_7
inst✝³ : Monoid G
inst✝² : AddCommMonoid M
inst✝¹ : MulAction G α
inst✝ : AddCommMonoid k
g : G
f f' : SkewMonoidAlgebra k G
⊢ g • (f + f') = g • f + g • f'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.UnitaryStarAlgAut

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : StarMul R
inst✝⁶ : CommRing S
inst✝⁵ : StarMul S
inst✝⁴ : Algebra S R
inst✝³ : StarModule S R
inst✝² : Algebra.IsCentral S R
inst✝¹ : IsCancelMulZero S
inst✝ : Module.IsTorsionFree S R
u v : ↥(unitary R)
x✝ : ∃ y, y • 1 = star ↑v * ↑u
y : S
h : y • 1 = star ↑v * ↑u
huv : ↑u = y • ↑v
hvu : ↑v = star y • ↑u
this : ↑v = 0
⊢ u = 1 • v

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝³ : Semiring k
inst✝² : Mul G
inst✝¹ : SMulZeroClass G k
inst✝ : DecidableEq G
f g : SkewMonoidAlgebra k G
x : G
⊢ (f * g).coeff x = f.sum fun a₁ b₁ ↦ g.sum fun a₂ b₂ ↦ if a₁ * a₂ = x then b₁ * a₁ • b₂ else 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : AddMonoidWithOne α
inst✝¹ : Invertible 2
a : α
inst✝ : Invertible a
⊢ sym ⅟a * sym a = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Star.UnitaryStarAlgAut

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : StarMul R
inst✝⁶ : CommRing S
inst✝⁵ : StarMul S
inst✝⁴ : Algebra S R
inst✝³ : StarModule S R
inst✝² : Algebra.IsCentral S R
inst✝¹ : IsCancelMulZero S
inst✝ : Module.IsTorsionFree S R
u v : ↥(unitary R)
⊢ (conjStarAlgAut S R) u = (conjStarAlgAut S R) v ↔ ∃ α, u = α • v

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
s : Finset (G × G)
hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x
F : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0
⊢ ∑ a₁ ∈ f.support, ∑ a₂ ∈ g.support, F (a₁, a₂) = ∑ p ∈ f.support ×ˢ g.support, F p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝³ : Mul α
inst✝² : AddMonoidWithOne α
inst✝¹ : Invertible 2
a : α
inst✝ : Invertible a
⊢ sym a * sym ⅟a = 1

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
s : Finset (G × G)
hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x
F : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0
⊢ {p ∈ f.support ×ˢ g.support | p.1 * p.2 = x} = {p ∈ s | p.1 ∈ f.support ∧ p.2 ∈ g.support}

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.TrivSqZeroExt.Ideal

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
r : R
m : M
hr : r = 0
⊢ (r, m) = inr (snd (r, m))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.TrivSqZeroExt.Ideal

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
r : R
m : M
hrm : (r, m) = inr (snd (r, m))
⊢ r = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.TrivSqZeroExt.Ideal

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
x : TrivSqZeroExt R M
⊢ x ∈ kerIdeal R M ↔ x = inr x.snd

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.TrivSqZeroExt.Ideal

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module Rᵐᵒᵖ M
inst✝ : IsCentralScalar R M
⊢ kerIdeal R M ^ 2 = ⊥

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
a b : Tropical R
⊢ untrop (a + b) = untrop (if a ≤ b then a else b)

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
a b : Tropical R
⊢ untrop (trop (max (untrop a) (untrop b))) = untrop (if a ≤ b then b else a)

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
x y : Tropical R
h : x ≤ y
⊢ untrop (x + y) = untrop x

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
x y : Tropical R
h : y ≤ x
⊢ untrop (x + y) = untrop y

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
x y : Tropical R
⊢ x + y = x ↔ x ≤ y

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
x y : Tropical R
⊢ x + y = y ↔ y ≤ x

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
s : Finset (G × G)
hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x
F : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0
p : G × G
hps : p ∈ s
hp : p ∉ {p ∈ s | p.1 ∈ f.support ∧ p.2 ∈ g.support}
⊢ f.coeff p.1 * p.1 • g.coeff p.2 = 0

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
x y z : Tropical R
⊢ x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
s : Finset (G × G)
hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x
⊢ (f * g).coeff x = ∑ p ∈ s, f.coeff p.1 * p.1 • g.coeff p.2

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
⊢ ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
a b : Tropical (WithTop R)
⊢ a + b = 0 ↔ a = 0 ∧ b = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
a b c : αˢʸᵐ
⊢ a * (b + c) = a * b + a * c

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝² : LinearOrder R
inst✝¹ : OrderTop R
inst✝ : Zero R
n : ℕ
⊢ untrop (if n + 1 = 0 then 0 else 1) = untrop ((if n = 0 then 0 else 1) + 1)

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : AddMonoid R
x✝ : Tropical R
⊢ untrop (x✝ ^ 0) = untrop 1

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : AddGroup R
x✝¹ x✝ : Tropical R
⊢ untrop (x✝¹ / x✝) = untrop (x✝¹ * x✝⁻¹)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
this : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite
s : Finset (G × G) := this.toFinset
F : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0
⊢ ∑ a₁ ∈ f.support, ∑ a₂ ∈ g.support, F (a₁, a₂) = ∑ p ∈ f.support ×ˢ g.support, F p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
a b c : αˢʸᵐ
⊢ (a + b) * c = a * c + b * c

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrder R
x y z : Tropical R
h : y ≤ z
⊢ x + y ≤ x + z

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
x✝ : αˢʸᵐ
⊢ 0 * x✝ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
x✝ : αˢʸᵐ
⊢ x✝ * 0 = 0

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
this : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite
s : Finset (G × G) := this.toFinset
F : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0
⊢ ∑ p ∈ f.support ×ˢ g.support with p.1 * p.2 = x, f.coeff p.1 * p.1 • g.coeff p.2 =
    ∑ p ∈ s with p.1 ∈ f.support ∧ p.2 ∈ g.support, f.coeff p.1 * p.1 • g.coeff p.2

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝³ : LinearOrder R
inst✝² : AddMonoid R
inst✝¹ : AddLeftMono R
inst✝ : AddRightMono R
x y : Tropical R
n : ℕ
⊢ (x + y) ^ n = x ^ n + y ^ n

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrderedAddCommMonoidWithTop R
x✝ : Tropical R
⊢ untrop (0 * x✝) = untrop 0

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : LinearOrderedAddCommMonoidWithTop R
x✝ : Tropical R
⊢ untrop (x✝ * 0) = untrop 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
x✝ : αˢʸᵐ
⊢ 1 * x✝ = x✝

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
n : ℕ
⊢ (n + 1) • x = x

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : AddCommMonoid R
a b : Tropical (WithTop R)
⊢ a * b = 0 ↔ a = 0 ∨ b = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
x✝ : αˢʸᵐ
⊢ x✝ * 1 = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
a : αˢʸᵐ
⊢ unsym (a * a) = unsym a * unsym a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : Invertible 2
a : α
⊢ sym (a * a) = sym a * sym a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝ : AddMonoid R
l : List R
⊢ trop l.sum = (map trop l).prod

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝ : AddCommMonoid R
s : Multiset R
⊢ ∀ (a : List R), trop (sum ⟦a⟧) = (map trop ⟦a⟧).prod

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝ : AddCommMonoid R
s : Finset S
f : S → R
⊢ trop (∑ i ∈ s, f i) = ∏ i ∈ s, trop (f i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝ : AddMonoid R
l : List (Tropical R)
⊢ untrop l.prod = (map untrop l).sum

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝ : AddCommMonoid R
s : Multiset (Tropical R)
⊢ ∀ (a : List (Tropical R)), untrop (prod ⟦a⟧) = (map untrop ⟦a⟧).sum

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝ : AddCommMonoid R
s : Finset S
f : S → Tropical R
⊢ untrop (∏ i ∈ s, f i) = ∑ i ∈ s, untrop (f i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝ : LinearOrder R
l : List R
⊢ trop l.minimum = (map (trop ∘ WithTop.some) l).sum

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝⁴ : Mul α
inst✝³ : AddCommSemigroup α
inst✝² : One α
inst✝¹ : OfNat α 2
inst✝ : Invertible 2
a b : αˢʸᵐ
⊢ a * b = b * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
s : Multiset R
⊢ trop s.inf = (map trop s).sum

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a✝ b : αˢʸᵐ
a : α
⊢ ?m.16 a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
s : Finset S
f : S → R
⊢ trop (s.inf f) = ∑ i ∈ s, trop (f i)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
this : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite
s : Finset (G × G) := this.toFinset
F : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0
p : G × G
hps : p ∈ s
hp : p ∉ {p ∈ s | p.1 ∈ f.support ∧ p.2 ∈ g.support}
⊢ f.coeff p.1 * p.1 • g.coeff p.2 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → WithTop R
⊢ trop (sInf (f '' ↑s)) = ∑ i ∈ s, trop (f i)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f g : SkewMonoidAlgebra k G
x : G
⊢ (f * g).coeff x = ∑ᶠ (p : G × G) (_ : p ∈ {p | p.1 * p.2 = x}), f.coeff p.1 * p.1 • g.coeff p.2

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), a * x = z ↔ a = y
a₁ : G
b₁ : k
⊢ (if a₁ * x = z then b₁ * a₁ • 0 else 0) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrder R
inst✝ : Fintype S
f : S → WithTop R
⊢ trop (⨅ i, f i) = ∑ i, trop (f i)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), a * x = z ↔ a = y
A :
  ∀ (a₁ : G) (b₁ : k),
    ((single x r).sum fun a₂ b₂ ↦ if a₁ * a₂ = z then b₁ * a₁ • b₂ else 0) = if a₁ * x = z then b₁ * a₁ • r else 0
⊢ (f * single x r).coeff z = f.sum fun a b ↦ if a = y then b * y • r else 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
s : Multiset (Tropical R)
⊢ untrop s.sum = (map untrop s).inf

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
s : Finset S
f : S → Tropical R
⊢ untrop (∑ i ∈ s, f i) = s.inf (untrop ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' ↑s)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), a * x = z ↔ a = y
A :
  ∀ (a₁ : G) (b₁ : k),
    ((single x r).sum fun a₂ b₂ ↦ if a₁ * a₂ = z then b₁ * a₁ • b₂ else 0) = if a₁ * x = z then b₁ * a₁ • r else 0
⊢ (if y ∈ f.support then f.coeff y * y • r else 0) = f.coeff y * y • r

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), a * x = z ↔ a = y
⊢ (f * single x r).coeff z = f.coeff y * y • r

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrder R
inst✝ : Fintype S
f : S → Tropical (WithTop R)
⊢ untrop (∑ i, f i) = ⨅ i, untrop (f i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Tropical.BigOperators

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ s, f i) = ⨅ i, untrop (f ↑i)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
r : k
g g' : G
x : SkewMonoidAlgebra k G
h : ∀ (x : G), ¬g' = x * g
⊢ (x * single g r).coeff g' = 0

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), x * a = y ↔ a = z
⊢ (f.sum fun a b ↦ if x * a = y then 0 * x • b else 0) = 0

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), x * a = y ↔ a = z
this : (f.sum fun a b ↦ if x * a = y then 0 * x • b else 0) = 0
⊢ (f.sum fun a b ↦ if x * a = y then r * x • b else 0) = f.sum fun a b ↦ if a = z then r * x • b else 0

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), x * a = y ↔ a = z
this : (f.sum fun a b ↦ if x * a = y then 0 * x • b else 0) = 0
⊢ (if z ∈ f.support then r * x • f.coeff z else 0) = r * x • f.coeff z

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
f : SkewMonoidAlgebra k G
r : k
x y z : G
H : ∀ (a : G), x * a = y ↔ a = z
⊢ (single x r * f).coeff y = r * x • f.coeff z

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Mul G
inst✝ : SMulZeroClass G k
r : k
g g' : G
x : SkewMonoidAlgebra k G
h : ¬∃ d, g' = g * d
⊢ (single g r * x).coeff g' = 0

Tactic:
by

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

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝² : Semiring k
inst✝¹ : Monoid G
inst✝ : MulSemiringAction G k
f : SkewMonoidAlgebra k G
r : k
x a : G
⊢ a * 1 = x ↔ a = x

Tactic:
by

Goal state after: