mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
a b : Γ
r s : R
x : Γ
h : ¬x = a + b
⊢ 0 = 0

pp: case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
a b : Γ
r s : R
x : Γ
h : ¬x = a + b
⊢ ∀ x_1 ∈ antidiagonal ⋯ ⋯ x, ((single a) r).coeff x_1.1 * ((single b) s).coeff x_1.2 = 0

Tactic:
]

Goal state after:
case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
a b : Γ
r s : R
x : Γ
h : ¬x = a + b
⊢ ∀ x_1 ∈ antidiagonal ⋯ ⋯ x, ((single a) r).coeff x_1.1 * ((single b) s).coeff x_1.2 = 0
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
f : Partial seed
x : ↥(↑f).domain
c : FiniteArchimedeanClass M
hx : ↑x ∈ seed.stratum c
hx' : ↑x ∈ seed.baseEmbedding.domain
heq : ↑⟨↑x, hx'⟩ = ↑x
⊢ ↑seed.baseEmbedding ⟨↑x, hx'⟩ = toLex ((single c) ((seed.coeff c) ⟨↑x, hx⟩))

Tactic:
]

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
f : Partial seed
x : ↥(↑f).domain
c : FiniteArchimedeanClass M
hx : ↑x ∈ seed.stratum c
hx' : ↑x ∈ seed.baseEmbedding.domain
heq : ↑⟨↑x, hx'⟩ = ↑x
⊢ ↑seed.baseEmbedding ⟨↑x, hx'⟩ = toLex ((single c) ((seed.coeff c) ⟨↑x, hx⟩))
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
a b : Γ
r s : R
y z : Γ
hy : (y, z).1 ∈ ((single a) r).support
hz : (y, z).2 ∈ ((single b) s).support
h : ¬a + b = a + b
⊢ ((single a) r).coeff (y, z).1 * ((single b) s).coeff (y, z).2 = 0

Tactic:
]

Goal state after:
case neg
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
a b : Γ
r s : R
y z : Γ
hy : (y, z).1 ∈ ((single a) r).support
hz : (y, z).2 ∈ ((single b) s).support
h : ¬a + b = a + b
⊢ ((single a) r).coeff (y, z).1 * ((single b) s).coeff (y, z).2 = 0
Module: Mathlib.Order.Quotient

Goal state before:
pp: case inl.inr
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≤ y
h₂ : y ≈ z
⊢ z < x → x ≈ z

Tactic:
]

Goal state after:
case inl.inr
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≤ y
h₂ : y ≈ z
⊢ z < x → x ≈ z
Module: Mathlib.Order.Quotient

Goal state before:
pp: case inl.inr
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≤ y
h₂ : ⟦y⟧ = ⟦z⟧
⊢ z < x → ⟦x⟧ = ⟦z⟧

Tactic:
]

Goal state after:
case inl.inr
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≤ y
h₂ : ⟦y⟧ = ⟦z⟧
⊢ z < x → ⟦x⟧ = ⟦z⟧
Module: Mathlib.Order.Quotient

Goal state before:
pp: case inr.inl
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≈ y
h₂ : y ≤ z
⊢ z < x → x ≈ z

Tactic:
]

Goal state after:
case inr.inl
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≈ y
h₂ : y ≤ z
⊢ z < x → x ≈ z
Module: Mathlib.Order.Quotient

Goal state before:
pp: case inr.inl
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : ⟦x⟧ = ⟦y⟧
h₂ : y ≤ z
⊢ z < x → ⟦x⟧ = ⟦z⟧

Tactic:
]

Goal state after:
case inr.inl
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : ⟦x⟧ = ⟦y⟧
h₂ : y ≤ z
⊢ z < x → ⟦x⟧ = ⟦z⟧
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case succ
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : Semiring R
a : Γ
r : R
n : ℕ
IH : (single a) r ^ n = (single (n • a)) (r ^ n)
⊢ (single (n • a + a)) (r ^ n * r) = (single (n • a + a)) (r ^ n * r)

Tactic:
]

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

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonAssocSemiring R
x y : R
⊢ (single 0) (x * y) = (single 0) (x * y)

Tactic:
]

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
f : Partial seed
x : ↥(↑f).domain
c : FiniteArchimedeanClass M
hx : ↑x ∈ seed.stratum c
hx' : ↑x ∈ seed.baseEmbedding.domain
heq : ↑⟨↑x, hx'⟩ = ↑x
fx : Π₀ (c : FiniteArchimedeanClass M), ↥(seed.stratum c) := DFinsupp.single c ⟨↑x, hx⟩
hfx : ↑x = fx.sum fun c ↦ ⇑(seed.stratum c).subtype
⊢ ofLex (↑seed.baseEmbedding ⟨↑x, hx'⟩) = (single c) ((seed.coeff c) ⟨↑x, hx⟩)

Tactic:
]

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
f : Partial seed
x : ↥(↑f).domain
c : FiniteArchimedeanClass M
hx : ↑x ∈ seed.stratum c
hx' : ↑x ∈ seed.baseEmbedding.domain
heq : ↑⟨↑x, hx'⟩ = ↑x
fx : Π₀ (c : FiniteArchimedeanClass M), ↥(seed.stratum c) := DFinsupp.single c ⟨↑x, hx⟩
hfx : ↑x = fx.sum fun c ↦ ⇑(seed.stratum c).subtype
⊢ ofLex (↑seed.baseEmbedding ⟨↑x, hx'⟩) = (single c) ((seed.coeff c) ⟨↑x, hx⟩)
Module: Mathlib.Order.Quotient

Goal state before:
pp: case h.h
α : Type u_1
s : Setoid α
inst✝¹ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
inst✝ : DecidableRel fun x1 x2 ↦ x1 ≈ x2
x y : α
h₁ : x ≤ y ∨ x ≈ y
h₂ : y ≤ x ∨ y ≈ x
⊢ ⟦x⟧ = ⟦y⟧

Tactic:
]

Goal state after:
case h.h
α : Type u_1
s : Setoid α
inst✝¹ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
inst✝ : DecidableRel fun x1 x2 ↦ x1 ≈ x2
x y : α
h₁ : x ≤ y ∨ x ≈ y
h₂ : y ≤ x ∨ y ≈ x
⊢ ⟦x⟧ = ⟦y⟧
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonAssocSemiring R
r s : R
rs : ∀ (x : Γ), (C r).coeff x = (C s).coeff x
⊢ r = s

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonAssocSemiring R
r s : R
rs : ∀ (x : Γ), (C r).coeff x = (C s).coeff x
⊢ r = s
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonAssocSemiring R
r s : R
rs : ∀ (x : Γ), (C r).coeff x = (C s).coeff x
h : r = s
⊢ r = s

Tactic:
]

Goal state after:
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonAssocSemiring R
r s : R
rs : ∀ (x : Γ), (C r).coeff x = (C s).coeff x
h : r = s
⊢ r = s
Module: Mathlib.Order.Quotient

Goal state before:
pp: α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y : α
⊢ y ≤ x ∨ x ≈ y ↔ y ≤ x ∨ x ≈ y

Tactic:
]

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

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonAssocSemiring R
r : R
h : r = 0
⊢ 0 = 0

Tactic:
]

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
f : Partial seed
x : ↥(↑f).domain
c : FiniteArchimedeanClass M
hx : ↑x ∈ seed.stratum c
hx' : ↑x ∈ seed.baseEmbedding.domain
heq : ↑⟨↑x, hx'⟩ = ↑x
fx : Π₀ (c : FiniteArchimedeanClass M), ↥(seed.stratum c) := DFinsupp.single c ⟨↑x, hx⟩
hfx : ↑x = fx.sum fun c ↦ ⇑(seed.stratum c).subtype
d : FiniteArchimedeanClass M
⊢ (seed.coeff d) (fx d) = ((single c) ((seed.coeff c) ⟨↑x, hx⟩)).coeff d

Tactic:
]

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
f : Partial seed
x : ↥(↑f).domain
c : FiniteArchimedeanClass M
hx : ↑x ∈ seed.stratum c
hx' : ↑x ∈ seed.baseEmbedding.domain
heq : ↑⟨↑x, hx'⟩ = ↑x
fx : Π₀ (c : FiniteArchimedeanClass M), ↥(seed.stratum c) := DFinsupp.single c ⟨↑x, hx⟩
hfx : ↑x = fx.sum fun c ↦ ⇑(seed.stratum c).subtype
d : FiniteArchimedeanClass M
⊢ (seed.coeff d) (fx d) = ((single c) ((seed.coeff c) ⟨↑x, hx⟩)).coeff d
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case neg
Γ : Type u_1
R : Type u_3
inst✝⁶ : AddCommMonoid Γ
inst✝⁵ : PartialOrder Γ
inst✝⁴ : IsOrderedCancelAddMonoid Γ
Γ' : Type u_6
inst✝³ : AddCommMonoid Γ'
inst✝² : PartialOrder Γ'
inst✝¹ : IsOrderedCancelAddMonoid Γ'
inst✝ : NonUnitalNonAssocSemiring R
f : Γ ↪o Γ'
hf : ∀ (x y : Γ), f (x + y) = f x + f y
x y : R⟦Γ⟧
g : Γ'
hg : g ∉ Set.range ⇑f
⊢ (embDomain f x * embDomain f y).coeff g = 0

Tactic:
]

Goal state after:
case neg
Γ : Type u_1
R : Type u_3
inst✝⁶ : AddCommMonoid Γ
inst✝⁵ : PartialOrder Γ
inst✝⁴ : IsOrderedCancelAddMonoid Γ
Γ' : Type u_6
inst✝³ : AddCommMonoid Γ'
inst✝² : PartialOrder Γ'
inst✝¹ : IsOrderedCancelAddMonoid Γ'
inst✝ : NonUnitalNonAssocSemiring R
f : Γ ↪o Γ'
hf : ∀ (x y : Γ), f (x + y) = f x + f y
x y : R⟦Γ⟧
g : Γ'
hg : g ∉ Set.range ⇑f
⊢ (embDomain f x * embDomain f y).coeff g = 0
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: case inr
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x y : ↥(↑f).domain
hnontrivial : Nontrivial M
hnonempty : Nonempty (FiniteArchimedeanClass M)
c : FiniteArchimedeanClass M
this : Nontrivial R
⊢ (ofLex (↑↑f x)).orderTop = (ofLex (↑↑f y)).orderTop ↔ ArchimedeanClass.mk (↑↑f x) = ArchimedeanClass.mk (↑↑f y)

Tactic:
]

Goal state after:
case inr
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x y : ↥(↑f).domain
hnontrivial : Nontrivial M
hnonempty : Nonempty (FiniteArchimedeanClass M)
c : FiniteArchimedeanClass M
this : Nontrivial R
⊢ (ofLex (↑↑f x)).orderTop = (ofLex (↑↑f y)).orderTop ↔ ArchimedeanClass.mk (↑↑f x) = ArchimedeanClass.mk (↑↑f y)
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : PartialOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : CommSemiring R
A : Type u_6
inst✝¹ : Semiring A
inst✝ : Algebra R A
r : R
x : A⟦Γ⟧
x✝ : Γ
⊢ (algebraMap R A) r * x.coeff x✝ = (algebraMap R A) r * x.coeff x✝

Tactic:
]

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

Goal state before:
pp: case inr
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x y : ↥(↑f).domain
hnontrivial : Nontrivial M
hnonempty : Nonempty (FiniteArchimedeanClass M)
c : FiniteArchimedeanClass M
this : Nontrivial R
⊢ ArchimedeanClass.mk (↑↑f x) = ArchimedeanClass.mk (↑↑f y) ↔ ArchimedeanClass.mk (↑↑f x) = ArchimedeanClass.mk (↑↑f y)

Tactic:
]

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

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
⊢ ∃ x, ¬(x ∈ ⊥ ↔ x ∈ ⊤)

Tactic:
]

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
⊢ ∃ x, ¬(x ∈ ⊥ ↔ x ∈ ⊤)
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
a : Γ
ha : a ≠ 0
x : R
⊢ ∃ x_1, ¬((algebraMap R R⟦Γ⟧) x).coeff x_1 = ((single a) 1).coeff x_1

Tactic:
]

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
a : Γ
ha : a ≠ 0
x : R
⊢ ∃ x_1, ¬((algebraMap R R⟦Γ⟧) x).coeff x_1 = ((single a) 1).coeff x_1
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
a : Γ
ha : a ≠ 0
x : R
⊢ ¬0 = 1

Tactic:
]

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁷ : AddCommMonoid Γ
inst✝⁶ : PartialOrder Γ
inst✝⁵ : IsOrderedCancelAddMonoid Γ
inst✝⁴ : CommSemiring R
A : Type u_6
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : Nontrivial Γ
inst✝ : Nontrivial R
a : Γ
ha : a ≠ 0
x : R
⊢ ¬0 = 1
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : ↥(↑f).domain
hx0 : ¬x = 0
hx0' : ↑x ≠ 0
x' : M
hx'mem : x' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
hx'0 : ⟨x', hx'mem⟩ ≠ 0
heq : ArchimedeanClass.mk x' = ArchimedeanClass.mk ↑x
x'' : ↥(↑f).domain := ⟨x', ⋯⟩
hx''mem : ↑x'' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
⊢ ⟨↑x'', hx''mem⟩ ≠ 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : ↥(↑f).domain
hx0 : ¬x = 0
hx0' : ↑x ≠ 0
x' : M
hx'mem : x' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
hx'0 : ⟨x', hx'mem⟩ ≠ 0
heq : ArchimedeanClass.mk x' = ArchimedeanClass.mk ↑x
x'' : ↥(↑f).domain := ⟨x', ⋯⟩
hx''mem : ↑x'' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
⊢ ⟨↑x'', hx''mem⟩ ≠ 0
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: case neg
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : ↥(↑f).domain
hx0 : ¬x = 0
hx0' : ↑x ≠ 0
x' : M
hx'mem : x' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
hx'0 : ⟨x', hx'mem⟩ ≠ 0
heq : ArchimedeanClass.mk x' = ArchimedeanClass.mk ↑x
x'' : ↥(↑f).domain := ⟨x', ⋯⟩
hx''mem : ↑x'' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
h0 : (seed.coeff (FiniteArchimedeanClass.mk (↑x) hx0')) ⟨↑x'', hx''mem⟩ ≠ 0
heq' : ↑(FiniteArchimedeanClass.mk (↑x) hx0') = (ofLex (↑↑f x)).orderTop
⊢ (withTopOrderIso M) (ofLex (↑↑f x)).orderTop = ArchimedeanClass.mk ↑x

Tactic:
]

Goal state after:
case neg
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : ↥(↑f).domain
hx0 : ¬x = 0
hx0' : ↑x ≠ 0
x' : M
hx'mem : x' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
hx'0 : ⟨x', hx'mem⟩ ≠ 0
heq : ArchimedeanClass.mk x' = ArchimedeanClass.mk ↑x
x'' : ↥(↑f).domain := ⟨x', ⋯⟩
hx''mem : ↑x'' ∈ seed.stratum (FiniteArchimedeanClass.mk (↑x) hx0')
h0 : (seed.coeff (FiniteArchimedeanClass.mk (↑x) hx0')) ⟨↑x'', hx''mem⟩ ≠ 0
heq' : ↑(FiniteArchimedeanClass.mk (↑x) hx0') = (ofLex (↑↑f x)).orderTop
⊢ (withTopOrderIso M) (ofLex (↑↑f x)).orderTop = ArchimedeanClass.mk ↑x
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case hfg
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : LinearOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : IsCancelAdd R
inst✝ : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this✝ : AddCancelCommMonoid R := { toAddCommMonoid := inst✝².toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ x, ¬y.coeff x = z.coeff x
this : {a | y.coeff a ≠ z.coeff a}.IsWF
a : Γ := this.min hyz
ha : y.coeff a ≠ z.coeff a
b c : Γ
hxb : ¬x.coeff b = 0
hbc : b + c = x.order + a
hbc' : y.coeff c ≠ z.coeff c
⊢ b + c = x.order + this.min hyz

Tactic:
]

Goal state after:
case hfg
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝⁵ : AddCommMonoid Γ
inst✝⁴ : LinearOrder Γ
inst✝³ : IsOrderedCancelAddMonoid Γ
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : IsCancelAdd R
inst✝ : IsCancelMulZero R
x : R⟦Γ⟧
hx : x ≠ 0
y z : R⟦Γ⟧
this✝ : AddCancelCommMonoid R := { toAddCommMonoid := inst✝².toAddCommMonoid, toIsLeftCancelAdd := ⋯ }
hyz : ∃ x, ¬y.coeff x = z.coeff x
this : {a | y.coeff a ≠ z.coeff a}.IsWF
a : Γ := this.min hyz
ha : y.coeff a ≠ z.coeff a
b c : Γ
hxb : ¬x.coeff b = 0
hbc : b + c = x.order + a
hbc' : y.coeff c ≠ z.coeff c
⊢ b + c = x.order + this.min hyz
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
c : FiniteArchimedeanClass M
x : ↥(↑f).domain
hx : ↑x ∈ ball K c
d : FiniteArchimedeanClass M
hd : ↑d ≤ ↑c
ne : x ≠ 0
⊢ ↑d < ArchimedeanClass.mk ↑x

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
c : FiniteArchimedeanClass M
x : ↥(↑f).domain
hx : ↑x ∈ ball K c
d : FiniteArchimedeanClass M
hd : ↑d ≤ ↑c
ne : x ≠ 0
⊢ ↑d < ArchimedeanClass.mk ↑x
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
y z : ↥(↑f).domain
c : FiniteArchimedeanClass M
hy : ↑y - x ∈ ball K c
hz : ↑z - x ∈ ball K c
d : FiniteArchimedeanClass M
hd : d ≤ c
⊢ (ofLex (↑↑f (y - z))).coeff d = 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
y z : ↥(↑f).domain
c : FiniteArchimedeanClass M
hy : ↑y - x ∈ ball K c
hz : ↑z - x ∈ ball K c
d : FiniteArchimedeanClass M
hd : d ≤ c
⊢ (ofLex (↑↑f (y - z))).coeff d = 0