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

Goal state before:
pp: case refine_1
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : gh ∈ antidiagonal ⋯ ⋯ x✝
⊢ gh ∈ antidiagonal ⋯ ⋯ x✝

pp: case refine_2
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : gh ∈ antidiagonal ⋯ ⋯ x✝
hz : gh ∉ antidiagonal ⋯ ⋯ x✝
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0

Tactic:
·

Goal state after:
case refine_1
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : gh ∈ antidiagonal ⋯ ⋯ x✝
⊢ gh ∈ antidiagonal ⋯ ⋯ x✝
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case neg
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case neg.inl
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
w : β
h2 : w ∈ s
h3 : f (Multiset.map g s).sum ≤ f (g w)
h4 : f (g a + (Multiset.map g s).sum) ≤ f (g a)
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)

pp: case neg.inr
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
w : β
h2 : w ∈ s
h3 : f (Multiset.map g s).sum ≤ f (g w)
h4 : f (g a + (Multiset.map g s).sum) ≤ f (Multiset.map g s).sum
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)

Tactic:
·

Goal state after:
case neg.inl
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
w : β
h2 : w ∈ s
h3 : f (Multiset.map g s).sum ≤ f (g w)
h4 : f (g a + (Multiset.map g s).sum) ≤ f (g a)
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case refine_2
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : gh ∈ antidiagonal ⋯ ⋯ x✝
hz : gh ∉ antidiagonal ⋯ ⋯ x✝
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0

Tactic:
·

Goal state after:
case refine_2
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : gh ∈ antidiagonal ⋯ ⋯ x✝
hz : gh ∉ antidiagonal ⋯ ⋯ x✝
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case neg.inl
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
w : β
h2 : w ∈ s
h3 : f (Multiset.map g s).sum ≤ f (g w)
h4 : f (g a + (Multiset.map g s).sum) ≤ f (g a)
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)

Tactic:
exact .inl h4

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

Goal state before:
pp: case inl
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 : R
⊢ mk x ≤ mk (f 0) + mk x

pp: case inr
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 : R
y : S
hy : y ≠ 0
⊢ mk x ≤ mk (f y) + mk x

Tactic:
·

Goal state after:
case inl
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 : R
⊢ mk x ≤ mk (f 0) + mk x
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case neg.inr
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
w : β
h2 : w ∈ s
h3 : f (Multiset.map g s).sum ≤ f (g w)
h4 : f (g a + (Multiset.map g s).sum) ≤ f (Multiset.map g s).sum
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)

Tactic:
·

Goal state after:
case neg.inr
R : Type u_1
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
hna : IsNonarchimedean f
g : β → α
a : β
s : Multiset β
h : s ≠ 0 → ∃ b ∈ s, f (Multiset.map g s).sum ≤ f (g b)
hs : a ::ₘ s ≠ 0
h1 : ¬s = 0
w : β
h2 : w ∈ s
h3 : f (Multiset.map g s).sum ≤ f (g w)
h4 : f (g a + (Multiset.map g s).sum) ≤ f (Multiset.map g s).sum
⊢ f (g a + (Multiset.map g s).sum) ≤ f (g a) ∨ ∃ a_1 ∈ s, f (g a + (Multiset.map g s).sum) ≤ f (g a_1)
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case inr
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 : R
y : S
hy : y ≠ 0
⊢ mk x ≤ mk (f y) + mk x

Tactic:
·

Goal state after:
case inr
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 : R
y : S
hy : y ≠ 0
⊢ mk x ≤ mk (f y) + mk x
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case pos
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x✝
hz : ¬f (x.coeff gh.1) = 0 → f (y.coeff gh.2) = 0
h : f (x.coeff gh.1) = 0
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0

pp: case neg
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x✝
hz : ¬f (x.coeff gh.1) = 0 → f (y.coeff gh.2) = 0
h : ¬f (x.coeff gh.1) = 0
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0

Tactic:
·

Goal state after:
case pos
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x✝
hz : ¬f (x.coeff gh.1) = 0 → f (y.coeff gh.2) = 0
h : f (x.coeff gh.1) = 0
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case neg
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x✝
hz : ¬f (x.coeff gh.1) = 0 → f (y.coeff gh.2) = 0
h : ¬f (x.coeff gh.1) = 0
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0

Tactic:
·

Goal state after:
case neg
Γ : Type u_1
R : Type u_3
S : Type u_4
inst✝⁴ : AddCommMonoid Γ
inst✝³ : PartialOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NonUnitalNonAssocSemiring S
f : R →ₙ+* S
x y : R⟦Γ⟧
x✝ : Γ
gh : Γ × Γ
hgh : ¬x.coeff gh.1 = 0 ∧ ¬y.coeff gh.2 = 0 ∧ gh.1 + gh.2 = x✝
hz : ¬f (x.coeff gh.1) = 0 → f (y.coeff gh.2) = 0
h : ¬f (x.coeff gh.1) = 0
⊢ f (x.coeff gh.1) * f (y.coeff gh.2) = 0
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case inl
R : Type u_1
inst✝³ : Semiring R
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
f_zero : f 0 = 0
f_nonneg : ∀ (x : α), 0 ≤ f x
hna : IsNonarchimedean f
g : β → α
⊢ ∃ i, (∅.Nonempty → i ∈ ∅) ∧ f (∅.sum g) ≤ f (g i)

pp: case inr
R : Type u_1
inst✝³ : Semiring R
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
f_zero : f 0 = 0
f_nonneg : ∀ (x : α), 0 ≤ f x
hna : IsNonarchimedean f
g : β → α
t : Finset β
ht : t.Nonempty
⊢ ∃ i, (t.Nonempty → i ∈ t) ∧ f (t.sum g) ≤ f (g i)

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝³ : Semiring R
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
f_zero : f 0 = 0
f_nonneg : ∀ (x : α), 0 ≤ f x
hna : IsNonarchimedean f
g : β → α
⊢ ∃ i, (∅.Nonempty → i ∈ ∅) ∧ f (∅.sum g) ≤ f (g i)
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : Semiring R
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
f_zero : f 0 = 0
f_nonneg : ∀ (x : α), 0 ≤ f x
hna : IsNonarchimedean f
g : β → α
t : Finset β
ht : t.Nonempty
⊢ ∃ i, (t.Nonempty → i ∈ t) ∧ f (t.sum g) ≤ f (g i)

Tactic:
·

Goal state after:
case inr
R : Type u_1
inst✝³ : Semiring R
inst✝² : LinearOrder R
α : Type u_2
β : Type u_3
inst✝¹ : AddCommMonoid α
inst✝ : Nonempty β
f : α → R
f_zero : f 0 = 0
f_nonneg : ∀ (x : α), 0 ≤ f x
hna : IsNonarchimedean f
g : β → α
t : Finset β
ht : t.Nonempty
⊢ ∃ i, (t.Nonempty → i ∈ t) ∧ f (t.sum g) ≤ f (g i)
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case h
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 : R
hx : 0 < mk x
y : S
hy : 0 < y
⊢ mk (f y) < mk x

pp: case hpos
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 : R
hx : 0 < mk x
y : S
hy : 0 < y
⊢ 0 ≤ f y

Tactic:
·

Goal state after:
case h
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 : R
hx : 0 < mk x
y : S
hy : 0 < y
⊢ mk (f y) < mk x
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case hpos
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 : R
hx : 0 < mk x
y : S
hy : 0 < y
⊢ 0 ≤ f y

Tactic:
·

Goal state after:
case hpos
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 : R
hx : 0 < mk x
y : S
hy : 0 < y
⊢ 0 ≤ f y
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case h
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 : R
hx : 0 < mk x
y : S
hy : y < 0
⊢ mk (f y) < mk x

pp: case hneg
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 : R
hx : 0 < mk x
y : S
hy : y < 0
⊢ f y ≤ 0

Tactic:
·

Goal state after:
case h
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 : R
hx : 0 < mk x
y : S
hy : y < 0
⊢ mk (f y) < mk x
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case hneg
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 : R
hx : 0 < mk x
y : S
hy : y < 0
⊢ f y ≤ 0

Tactic:
·

Goal state after:
case hneg
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 : R
hx : 0 < mk x
y : S
hy : y < 0
⊢ f y ≤ 0
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : LinearOrder R
α : Type u_2
β : Type u_3
inst✝ : AddCommGroup α
f : α → R
fna : IsNonarchimedean f
f_neg : ∀ (a : α), f a = f (-a)
s : Finset β
l : β → α
k : β
hk : k ∈ s
hmax : ∀ j ∈ s, j ≠ k → f (l j) < f (l k)
hcard : s.card = 1
⊢ f (∑ i ∈ s, l i) = f (l k)

pp: case neg
R : Type u_1
inst✝¹ : LinearOrder R
α : Type u_2
β : Type u_3
inst✝ : AddCommGroup α
f : α → R
fna : IsNonarchimedean f
f_neg : ∀ (a : α), f a = f (-a)
s : Finset β
l : β → α
k : β
hk : k ∈ s
hmax : ∀ j ∈ s, j ≠ k → f (l j) < f (l k)
hcard : ¬s.card = 1
⊢ f (∑ i ∈ s, l i) = f (l k)

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝¹ : LinearOrder R
α : Type u_2
β : Type u_3
inst✝ : AddCommGroup α
f : α → R
fna : IsNonarchimedean f
f_neg : ∀ (a : α), f a = f (-a)
s : Finset β
l : β → α
k : β
hk : k ∈ s
hmax : ∀ j ∈ s, j ≠ k → f (l j) < f (l k)
hcard : s.card = 1
⊢ f (∑ i ∈ s, l i) = f (l k)
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: case pos
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)
hmono : StrictMonoOn (fun x ↦ ArchimedeanClass.mk ↑(f x)) ↑f.support
hsupport' : (f.support.erase (f.support.min' hsupport)).Nonempty
⊢ ↑(f.support.min' hsupport) < ArchimedeanClass.mk (∑ x ∈ f.support.erase (f.support.min' hsupport), ↑(f x))

pp: case neg
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)
hmono : StrictMonoOn (fun x ↦ ArchimedeanClass.mk ↑(f x)) ↑f.support
hsupport' : f.support.erase (f.support.min' hsupport) = ∅
⊢ ↑(f.support.min' hsupport) < ArchimedeanClass.mk (∑ x ∈ f.support.erase (f.support.min' hsupport), ↑(f x))

Tactic:
·

Goal state after:
case pos
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)
hmono : StrictMonoOn (fun x ↦ ArchimedeanClass.mk ↑(f x)) ↑f.support
hsupport' : (f.support.erase (f.support.min' hsupport)).Nonempty
⊢ ↑(f.support.min' hsupport) < ArchimedeanClass.mk (∑ x ∈ f.support.erase (f.support.min' hsupport), ↑(f x))
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✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : r = 0
⊢ (x * (single b) r).coeff (a + b) = x.coeff a * r

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
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
⊢ (x * (single b) r).coeff (a + b) = x.coeff a * r

Tactic:
·

Goal state after:
case pos
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : r = 0
⊢ (x * (single b) r).coeff (a + b) = x.coeff a * r
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : LinearOrder R
α : Type u_2
β : Type u_3
inst✝ : AddCommGroup α
f : α → R
fna : IsNonarchimedean f
f_neg : ∀ (a : α), f a = f (-a)
s : Finset β
l : β → α
k : β
hk : k ∈ s
hmax : ∀ j ∈ s, j ≠ k → f (l j) < f (l k)
hcard : ¬s.card = 1
⊢ f (∑ i ∈ s, l i) = f (l k)

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝¹ : LinearOrder R
α : Type u_2
β : Type u_3
inst✝ : AddCommGroup α
f : α → R
fna : IsNonarchimedean f
f_neg : ∀ (a : α), f a = f (-a)
s : Finset β
l : β → α
k : β
hk : k ∈ s
hmax : ∀ j ∈ s, j ≠ k → f (l j) < f (l k)
hcard : ¬s.card = 1
⊢ f (∑ i ∈ s, l i) = f (l k)
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case inl
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
h : n ≠ 0
h✝ : Subsingleton S
⊢ mk ↑n = 0

pp: case inr
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
h : n ≠ 0
h✝ : Nontrivial S
⊢ mk ↑n = 0

Tactic:
·

Goal state after:
case inl
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
h : n ≠ 0
h✝ : Subsingleton S
⊢ mk ↑n = 0
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case inr
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
h : n ≠ 0
h✝ : Nontrivial S
⊢ mk ↑n = 0

Tactic:
·

Goal state after:
case inr
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
h : n ≠ 0
h✝ : Nontrivial S
⊢ mk ↑n = 0
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✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : x.coeff a = 0
⊢ ∑ x_1 ∈ antidiagonal ⋯ ⋯ (a + b), x.coeff x_1.1 * ((single b) r).coeff x_1.2 = x.coeff a * r

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
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ ∑ x_1 ∈ antidiagonal ⋯ ⋯ (a + b), x.coeff x_1.1 * ((single b) r).coeff x_1.2 = x.coeff a * r

Tactic:
·

Goal state after:
case pos
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : x.coeff a = 0
⊢ ∑ x_1 ∈ antidiagonal ⋯ ⋯ (a + b), x.coeff x_1.1 * ((single b) r).coeff x_1.2 = x.coeff a * r
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case inl
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
⊢ 0 ≤ mk ↑0

pp: case inr
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
hn : n ≠ 0
⊢ 0 ≤ mk ↑n

Tactic:
·

Goal state after:
case inl
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
⊢ 0 ≤ mk ↑0
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: case inr
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
hn : n ≠ 0
⊢ 0 ≤ mk ↑n

Tactic:
·

Goal state after:
case inr
S : Type u_3
inst✝² : LinearOrder S
inst✝¹ : CommRing S
inst✝ : IsStrictOrderedRing S
n : ℤ
hn : n ≠ 0
⊢ 0 ≤ mk ↑n
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✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ ∑ x_1 ∈ antidiagonal ⋯ ⋯ (a + b), x.coeff x_1.1 * ((single b) r).coeff x_1.2 =
    ∑ ij ∈ {(a, b)}, x.coeff ij.1 * ((single b) r).coeff ij.2

pp: Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ ∑ ij ∈ {(a, b)}, x.coeff ij.1 * ((single b) r).coeff ij.2 = x.coeff a * r

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✝ : NonUnitalNonAssocSemiring R
r : R
x : R⟦Γ⟧
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ ∑ x_1 ∈ antidiagonal ⋯ ⋯ (a + b), x.coeff x_1.1 * ((single b) r).coeff x_1.2 =
    ∑ ij ∈ {(a, b)}, x.coeff ij.1 * ((single b) r).coeff ij.2